#7598041

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	#45269	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1504 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)(17,18,23,26)(19,22,25,21)(20,28,29,27)(24,31,32,30);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,22)(20,24)(21,26)(23,25)(27,30)(28,31)(29,32);(10,13)(14,15)(17,20,21,30,23,29,22,31)(18,24,25,28,26,32,19,27);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,14,13,15)(17,19,21,18,23,25,22,26)(20,27,30,24,29,28,31,32);(1,4,5,3)(2,7,6,8)(20,29)(24,32)(27,28)(30,31)

Polycyclic presentation:

2^9|a^2=i,c^2=hi,d^2=fg,e^2=f,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, 7505237);