#357427

order	512
nilpotency class	3
isoclinism	isoclinic class of #184767
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#535	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

5102 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,h^2=i,b^a=be,c^a=cf,e^a=ei,c^b=cg,h^b=hi,d^c=di

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, 264623);