#354708

order	512
nilpotency class	3
isoclinism	isoclinic class of #447
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₈	C₂×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1144 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=f,d^2=eg,e^2=i,b^a=be,c^a=cf,e^b=ei,g^b=gi

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261904);