#520732

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁶×C₄²

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	C₂²×#120	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

4296 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,e^2=fg,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.

Relationships

maximal quotient of

#72313643

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 427928);