#614415

order	512
nilpotency class	2
isoclinism	isoclinic class of #612424
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

4582 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=h,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72310199, #72310207, #72310221
maximal normal subgroup of	#72310199, #72310207, #72310221

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 521611);