#207308

order	512
nilpotency class	3
isoclinism	isoclinic class of #1999
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄²	C₂×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂²×C₄⋊C₄
prime radical	C₂⁴	C₂×C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

724 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=fh,e^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,c^b=ci,e^b=eh

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, 114504);