#7603026

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518944
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	#2431	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1427 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=h,d^2=fg,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cgi,d^b=dh,g^b=gi,h^b=hi,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, 7510222);