#130291

order	512
nilpotency class	4
isoclinism	isoclinic class of #130290
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	#62284	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

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

normal subgroups only

This group has 590 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240824, #72250769, #72250809, …
maximal normal subgroup of	#72240824, #72250769, #72250809, …
maximal quotient of	#72240799

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 37487);