#4866448

order	512
nilpotency class	2
isoclinism	isoclinic class of #4861280
automorphism	Aut(G) order 4194304
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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

1974 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(2,4)(5,6,8,10)(7,11,12,9)(13,14)(15,21)(16,24)(17,18)(19,23)(20,26)(22,27)(25,28)(30,32)(35,37)(36,40)(42,43);(7,12)(9,11)(14,18)(19,21)(20,24)(27,28)(29,30,34,32)(31,35,39,37)(33,36,41,40)(38,42,44,43);(5,6)(7,9)(8,10)(11,12)(13,14)(15,19)(16,20)(17,18)(21,23)(22,27)(24,26)(25,28)(29,31,34,39)(30,35,32,37)(33,38,41,44)(36,42,40,43);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(13,15,17,23)(14,19,18,21)(16,25,26,22)(20,28,24,27)(29,32,34,30)(31,37,39,35)(33,36,41,40)(38,42,44,43);(13,16,17,26)(14,20,18,24)(15,22,23,25)(19,27,21,28)(29,33)(30,36)(31,38)(32,40)(34,41)(35,42)(37,43)(39,44)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4773644);