#894286

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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

3310 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319866
maximal normal subgroup of	#72319866
maximal quotient of	#72320263

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 801482);