#6529934

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499823
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2433	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1836 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 36 points:

(2,4)(21,22)(23,31)(24,27)(25,29)(26,28)(30,34)(32,35)(33,36);(1,2,3,4)(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,23,25,24)(22,27,29,31)(26,32,34,33)(28,36,30,35);(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18)(22,28)(23,24)(26,34)(27,35)(29,30)(31,36);(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19)(21,24,25,23)(22,27,29,31)(26,33,34,32)(28,36,30,35);(6,12)(10,13)(11,20)(18,19)(23,24)(27,31)(32,33)(35,36)

Polycyclic presentation:

2^9|b^2=f,c^2=hi,d^2=h,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257144, #72257176, #72257213, …
maximal normal subgroup of	#72257144, #72257176, #72257213, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6437130);