#130236

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	#62330	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

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

normal subgroups only

This group has 597 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=hi,e^2=i,f^2=hi,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, #72250549, #72250556, …
maximal normal subgroup of	#72240824, #72250549, #72250556, …
maximal quotient of	#72240816

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 37432);