#385083

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

917 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)(9,10)(11,15)(12,14)(13,16)(17,18,22,26,21,25,31,32)(19,20,28,30,27,23,24,29);(1,2,4,5)(3,8,7,6)(9,11)(10,13)(12,16)(14,15)(17,19)(18,23)(20,25)(21,27)(22,28)(24,31)(26,29)(30,32);(10,14)(13,15)(17,20)(18,24)(19,26)(21,23)(22,30)(25,28)(27,32)(29,31);(1,3,4,7)(2,6,5,8)(11,16)(13,15)(18,25)(19,27)(24,28)(26,32)

Polycyclic presentation:

2^9|a^2=h,b^2=ef,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, 292279);