#7600941

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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1561 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)(4,5)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18,23,27)(19,28,26,22)(20,24,25,32)(21,30,31,29);(2,5)(3,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,18,23,26,28,27)(20,21,29,24,25,31,30,32);(1,3,4,8)(2,6,5,7)(10,14)(11,16)(17,20,23,25)(18,24,27,32)(19,21,26,31)(22,29,28,30);(2,5)(6,7)(10,14)(11,16)(20,25)(21,31)(24,32)(29,30);(2,5)(6,7)(10,14)(13,15)(17,21,22,24,23,31,28,32)(18,25,26,30,27,20,19,29)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=ghi,e^2=fg,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,d^c=di,e^c=ei,e^d=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, 7508137);