#7042356

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
prime radical	#45353	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1895 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)(5,6,9,12)(7,10,14,18)(8,11,16,20)(13,17,15,19)(21,22,28,30,27,23,26,25)(24,29,33,36,35,31,34,32);(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(21,23,27,22)(24,31,35,29)(25,28,30,26)(32,33,36,34);(1,2,3,4)(5,8)(6,11)(7,13)(9,16)(10,17)(12,20)(14,15)(18,19)(21,24)(22,29)(23,31)(25,32)(26,34)(27,35)(28,33)(30,36);(8,16)(11,20)(13,15)(17,19)(21,25,26,23,27,30,28,22)(24,32,34,31,35,36,33,29);(6,12)(7,14)(11,20)(13,15)(21,26,27,28)(22,30,23,25)(24,33,35,34)(29,32,31,36)

Polycyclic presentation:

2^9|a^2=fhi,b^2=i,c^2=g,d^2=f,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ehi,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6949552);