#410862

order	512
nilpotency class	3
isoclinism	isoclinic class of #393877
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	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1360 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=gh,c^2=g,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,e^b=ei,f^b=fi,e^c=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, 318058);