#7191168

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993915
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	#1999	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1739 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)(21,22,27,30)(23,31,35,36)(24,25,29,26)(28,33,32,34);(5,7,10,6)(8,11,15,13)(9,12,17,14)(16,19,20,18)(21,23,26,34,27,35,25,33)(22,28,29,31,30,32,24,36);(2,4)(5,8)(6,11)(7,13)(9,16)(10,15)(12,18)(14,19)(17,20);(8,15)(11,13)(16,20)(18,19)(21,24)(22,25)(23,32)(26,30)(27,29)(28,35)(31,34)(33,36);(5,9,10,17)(6,12,7,14)(8,16,15,20)(11,18,13,19)(21,25,27,26)(22,29,30,24)(23,33,35,34)(28,31,32,36)

Polycyclic presentation:

2^9|a^2=hi,b^2=fh,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,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, 7098364);