#358737

order	512
nilpotency class	3
isoclinism	isoclinic class of #23387
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1114 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=e,d^2=ef,g^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^b=di,g^b=gi,g^c=gi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 265933);