#376241

order	512
nilpotency class	3
isoclinism	isoclinic class of #24283
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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	C₂³×D₈	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1520 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239138, #72239140, #72239148
maximal normal subgroup of	#72239138, #72239140, #72239148

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 283437);