#369053

order	512
nilpotency class	3
isoclinism	isoclinic class of #24404
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂×#2539	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

2722 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=fg,d^2=ei,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,f^b=fi,f^c=fi,g^c=gi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 276249);