#393174

order	512
nilpotency class	3
isoclinism	isoclinic class of #32944
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1332 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,c^2=fi,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,c^b=ci,g^d=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 300370);