#471988

order	512
nilpotency class	3
isoclinism	isoclinic class of #454375
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1606 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=egh,d^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de,f^c=fi,g^c=gi

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, 379184);