#6547164

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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	C₂×#3366	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1800 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308515, #72308845
maximal normal subgroup of	#72308515, #72308845

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6454360);