#387124

order	512
nilpotency class	3
isoclinism	isoclinic class of #30412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1128 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eg,c^2=g,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=dei,f^c=fi,f^d=fi

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, 294320);