#358487

order	512
nilpotency class	3
isoclinism	isoclinic class of #23400
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2154 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)(5,6,7,8)(9,10)(11,19)(12,16)(13,17)(14,18)(15,20)(21,23)(22,24);(9,11,12,20,14,21,13,22)(10,15,17,19,18,24,16,23);(1,2,3,4)(9,10)(11,19)(12,17)(13,16)(14,18)(15,22)(20,24)(21,23);(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,24,19)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=f,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,c^b=cg,g^b=gi,d^c=di,g^c=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, 265683);