#7173356

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996355
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

1765 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)(5,6,9,12)(7,10,13,17)(8,15,16,11)(14,20,19,18)(21,22)(23,29)(24,33)(25,27)(26,31)(28,34)(30,35)(32,36);(1,2)(3,4)(7,13)(8,16)(10,17)(11,15)(21,22)(23,30)(24,28)(25,27)(26,32)(29,35)(31,36)(33,34);(5,7)(6,10)(8,14)(9,13)(11,18)(12,17)(15,20)(16,19)(21,23,24,31,25,32,34,35)(22,26,28,29,27,30,33,36);(2,4)(7,13)(10,17)(14,19)(18,20)(21,22)(23,26)(24,28)(25,27)(29,31)(30,32)(33,34)(35,36);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(22,27)(26,30)(28,33)(29,36)

Polycyclic presentation:

2^9|a^2=hi,c^2=g,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,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, 7080552);