#7595997

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	#44892	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1867 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)(17,18,21,23)(19,27,30,20)(22,31,29,26)(24,25,32,28);(2,6)(7,8)(10,13)(11,16)(18,23)(19,28)(20,24)(22,29)(25,30)(27,32);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,14,13,15)(17,19,29,25,21,30,22,28)(18,24,31,20,23,32,26,27);(1,4,5,3)(2,8,6,7)(10,13)(14,15)(17,20,29,32,21,27,22,24)(18,25,31,30,23,28,26,19);(2,6)(7,8)(10,13)(14,15)(18,23)(20,27)(24,32)(26,31)

Polycyclic presentation:

2^9|a^2=i,c^2=fh,d^2=fg,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, 7503193);