#7185442

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996405
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#61209	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1723 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gi,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7092638);