#365022

order	512
nilpotency class	3
isoclinism	isoclinic class of #24292
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#472	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1780 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,d^2=ei,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gi,d^c=di,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226901, #72226917, #72226931
maximal normal subgroup of	#72226901, #72226917, #72226931

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 272218);