#7604808

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518992
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#125 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1275 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

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

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, 7512004);