#448388

order	512
nilpotency class	3
isoclinism	isoclinic class of #417508
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1391 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=ei,c^2=e,d^2=eg,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,h^c=hi,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 355584);