#7597454

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
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	#25761	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1642 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)(3,7)(4,8)(5,6)(9,10)(11,15)(12,13)(14,16)(17,18)(19,27)(20,22)(21,25)(23,28)(24,29)(26,30)(31,32);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,21,28)(18,23,25,27)(20,24,26,32)(22,31,30,29);(1,3,5,4)(2,7,6,8)(10,13)(14,15)(17,20,30,25,21,26,22,18)(19,23,31,32,28,27,29,24);(9,11,12,16)(10,14,13,15)(17,19,22,29,21,28,30,31)(18,24,26,27,25,32,20,23);(1,4,5,3)(2,7,6,8)(18,25)(20,26)(23,27)(24,32)

Polycyclic presentation:

2^9|b^2=i,c^2=fh,d^2=g,e^2=fi,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=cg,d^b=dh,g^b=gi,h^b=hi,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7504650);