#7181262

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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1724 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)(21,22,28,36,27,31,26,30)(23,32,25,24,29,34,33,35);(5,6,9,12)(7,13,14,10)(8,17,16,11)(15,18,20,19)(21,23,27,29)(22,24,31,35)(25,28,33,26)(30,34,36,32);(2,4)(5,7)(6,10)(8,15)(9,14)(11,18)(12,13)(16,20)(17,19)(21,24)(22,29)(23,31)(25,30)(26,32)(27,35)(28,34)(33,36);(7,14)(10,13)(15,20)(18,19)(21,25,26,23,27,33,28,29)(22,24,30,32,31,35,36,34);(5,8,9,16)(6,11,12,17)(7,15,14,20)(10,18,13,19)(21,26,27,28)(22,30,31,36)(23,25,29,33)(24,34,35,32)

Polycyclic presentation:

2^9|a^2=fi,b^2=hi,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7088458);