#370556

order	512
nilpotency class	2
isoclinism	isoclinic class of #14715
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

1366 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)(12,13)(14,16)(17,18,22,25,21,24,26,19)(20,23,30,28,29,31,32,27);(1,2,4,3)(5,6)(7,8)(10,13)(14,15)(17,19,26,24,21,25,22,18)(20,28,32,23,29,27,30,31);(1,2,4,3)(6,8)(17,20,21,29)(18,23,24,31)(19,27,25,28)(22,30,26,32);(1,3,4,2)(9,11,12,16)(10,14,13,15)(20,29)(23,31)(27,28)(30,32)

Polycyclic presentation:

2^9|a^2=h,b^2=fgh,c^2=fg,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^c=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 277752);