#7595564

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518936
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1651 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,14)(13,16)(17,18,22,25)(19,27,29,24)(20,21,26,23)(28,31,30,32);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,29)(18,24,25,27)(20,31,26,32)(21,28,23,30);(10,14)(11,16)(17,20,23,18,22,26,21,25)(19,27,28,32,29,24,30,31);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,20,23,18,22,26,21,25)(19,27,28,32,29,24,30,31);(2,5)(6,8)(10,14)(13,15)(17,21,22,23)(18,20,25,26)(19,28,29,30)(24,31,27,32)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=hi,d^2=fgi,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=cgi,d^b=dh,g^b=gi,h^b=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7502760);