#7595986

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	#44927	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1975 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

2^9|c^2=fhi,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=cg,d^b=dh,g^b=gi,h^b=hi,d^c=di,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, 7503182);