#377473

order	512
nilpotency class	3
isoclinism	isoclinic class of #24289
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2358
prime radical	#3449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1133 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 32

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72239933, #72239938, #72239940

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 284669);