#385008

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 ◁ C₂×#336 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

838 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=e,d^2=f,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, 292204);