#140394

order	512
nilpotency class	3
isoclinism	isoclinic class of #447
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₈	C₂×D₄
commutator	C₂×C₄	C₂²×C₁₆
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle, prime radical	C₂³	C₂×C₄⋊C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

278 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=eg,d^2=g,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,d^b=dg

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47590);