#520941

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂⁴
commutator	C₂³	C₂⁴×C₄
Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2878 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fg,d^2=fg,e^2=fh,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df

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, 428137);