#7188606

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994171
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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

1406 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2)(3,6)(4,5)(7,8)(12,20)(15,21)(19,24)(22,23)(25,26,30,33)(27,35,37,40)(28,34,29,32)(31,38,39,36);(2,5)(3,7)(9,10,13,16)(11,17,18,14)(12,21,20,15)(19,22,24,23)(25,27,29,36,30,37,28,38)(26,31,34,35,33,39,32,40);(2,5)(6,8)(9,11)(10,14)(12,19)(13,18)(15,22)(16,17)(20,24)(21,23)(25,28,30,29)(26,32,33,34)(27,36,37,38)(31,35,39,40);(1,3,4,7)(2,6,5,8)(11,18)(14,17)(19,24)(22,23)(27,36)(28,29)(31,35)(32,34)(37,38)(39,40);(9,12,13,20)(10,15,16,21)(11,19,18,24)(14,22,17,23)(26,33)(27,37)(32,34)(36,38)

Polycyclic presentation:

2^9|a^2=i,b^2=fgh,c^2=i,d^2=gi,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,f^d=fi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7095802);