#7183950

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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 ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1867 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=hi,c^2=i,d^2=f,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307373, #72307685, #72307818, …
maximal normal subgroup of	#72307373, #72307685, #72307818, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7091146);