#361355

order	512
nilpotency class	3
isoclinism	isoclinic class of #23848
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1548 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 268551);