#393330

order	512
nilpotency class	3
isoclinism	isoclinic class of #32935
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1982 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)(3,4)(5,6,7,8)(9,10)(11,15)(12,13)(14,16)(17,18)(19,23)(20,24)(21,22);(6,8)(10,13)(14,15)(17,19,23,18,21,24,20,22);(9,11,12,16)(10,14,13,15)(18,22)(19,24);(2,4)(18,22)(19,24)

Polycyclic presentation:

2^9|a^2=h,b^2=efh,c^2=fi,e^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,c^b=ci,d^b=di

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, 300526);