#7191207

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994043
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1783 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)(5,6,10,7)(8,11,15,13)(9,14,17,12)(16,19,20,18)(21,22,26,29)(23,30,32,36)(24,28,25,35)(27,34,31,33);(5,7,10,6)(8,11,15,13)(9,12,17,14)(16,19,20,18)(21,23,24,34,26,32,25,33)(22,27,35,36,29,31,28,30);(2,4)(5,8)(6,11)(7,13)(9,16)(10,15)(12,18)(14,19)(17,20)(21,22)(23,31)(24,35)(25,28)(26,29)(27,32)(30,34)(33,36);(8,15)(11,13)(16,20)(18,19)(21,24)(22,28)(23,32)(25,26)(29,35)(30,36);(5,9,10,17)(6,12,7,14)(8,16,15,20)(11,18,13,19)

Polycyclic presentation:

2^9|a^2=h,b^2=fh,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=df,e^b=ehi,d^c=dh,f^d=fi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7098403);