#130232

order	512
nilpotency class	4
isoclinism	isoclinic class of #130232
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#8350
commutator	C₂×M₁₆	C₂²×C₄
Frattini, cosocle	C₂×#131	C₂³
prime radical	#62327	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

599 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 599 conjugacy classes of subgroups — too many to draw. Only its 90 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,7,14,8,16,35,50,9,15,36,44,6,13,25,48)(3,17,21,53,22,43,34,38,23,54,61,64,20,52,60,63)(4,24,27,45,26,42,5,29,28,46,31,47,33,11,32,12)(10,37,40,19,39,18,55,30,41,56,62,58,51,57,49,59);(1,3)(2,10)(4,18)(5,30)(6,22)(7,34)(8,20)(9,23)(11,38)(12,43)(13,39)(14,49)(15,41)(16,51)(17,45)(19,31)(21,35)(24,53)(25,61)(26,37)(27,58)(28,57)(29,52)(32,59)(33,56)(36,60)(40,48)(42,63)(44,55)(46,64)(47,54)(50,62);(1,4)(2,11)(3,18)(5,25)(6,33)(7,31)(8,26)(9,28)(10,38)(12,44)(13,46)(14,29)(15,42)(16,24)(17,40)(19,34)(20,37)(21,59)(22,56)(23,57)(27,36)(30,61)(32,35)(39,64)(41,63)(43,55)(45,48)(47,50)(49,52)(51,53)(54,62)(58,60)

Polycyclic presentation:

2^9|a^2=f,d^2=hi,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dh,e^b=egi,f^b=fgh,h^b=hi,d^c=dg,e^c=ei,f^c=fi,e^d=ei,f^d=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240824, #72250520, #72250541, …
maximal normal subgroup of	#72240824, #72250520, #72250541, …
maximal quotient of	#72240800

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 37428);