#96879

order	512
nilpotency class	3
isoclinism	isoclinic class of #7654
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₄³	C₂³
socle, prime radical	C₂³	C₄×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

408 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=eg,d^2=g,e^2=i,f^2=gh,b^a=bd,c^a=ci,d^a=dg,f^a=fgh,c^b=ch,d^b=dh

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4075);