#6565787

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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1887 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)(5,6,10,9)(7,11,13,15)(8,12,16,17)(14,18,19,20)(21,22,25,28)(23,26,30,34)(24,27,32,29)(31,33,36,35);(1,2,3,4)(5,7)(6,11)(8,14)(9,15)(10,13)(12,18)(16,19)(17,20)(21,23)(22,26)(24,31)(25,30)(27,33)(28,34)(29,35)(32,36);(7,13)(11,15)(14,19)(18,20)(22,27)(24,32)(26,33)(28,29)(31,36)(34,35);(5,8,10,16)(6,12,9,17)(7,14,13,19)(11,18,15,20)(23,30)(26,34)(31,36)(33,35);(5,9,10,6)(7,15,13,11)(8,12,16,17)(14,18,19,20)(22,28)(26,34)(27,29)(33,35)

Polycyclic presentation:

2^9|a^2=hi,b^2=f,d^2=h,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,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	#72307339, #72307500, #72307734, …
maximal normal subgroup of	#72307339, #72307500, #72307734, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6472983);