#146834

order	512
nilpotency class	2
isoclinism	isoclinic class of #7855
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₄²
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂³	C₂×#97
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

552 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=d,d^2=i,f^2=h,g^2=h,h^2=i,b^a=bd,c^a=ce,g^a=gi,f^b=fi

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54030);