#609028

order	512
nilpotency class	2
isoclinism	isoclinic class of #15257
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

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

4160 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239781, #72280075, #72280103, …
maximal normal subgroup of	#72239781, #72280075, #72280103, …
maximal quotient of	#72256876, #72298426, C₂×#609028

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 516224);