#385774

order	512
nilpotency class	3
isoclinism	isoclinic class of #30412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂×C₄²

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	C₂×C₄×D₄
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄⋊C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

753 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)(10,13)(11,15)(17,18,23,28,21,26,22,27)(19,20,30,31,29,24,25,32);(1,2,4,5)(3,8,7,6)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,26)(21,29)(22,25)(23,30)(27,31)(28,32);(9,11,12,15)(10,14,13,16)(17,20,21,24)(18,25,26,30)(19,27,29,28)(22,32,23,31);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(18,26)(19,29)(25,30)(27,28)

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=f,d^2=ef,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,g^b=gi,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, 292970);