#4941293

order	512
nilpotency class	2
isoclinism	isoclinic class of #4932388
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

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

4685 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fh,d^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4848489);