#847113

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

4610 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,d^2=g,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 754309);