#4866489

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

1904 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)(13,14,18,15)(16,21,24,19)(17,22,26,20)(23,27,25,28)(30,34)(31,37)(36,39)(38,43);(7,12)(9,11)(14,15)(19,21)(20,22)(27,28)(29,30,33,34)(31,35,37,41)(32,36,40,39)(38,42,43,44);(5,6)(7,9)(8,10)(11,12)(13,15,18,14)(16,21,24,19)(17,22,26,20)(23,28,25,27)(29,31)(30,35)(32,38)(33,37)(34,41)(36,42)(39,44)(40,43);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(13,16,18,24)(14,19,15,21)(17,25,26,23)(20,28,22,27)(29,30)(31,35)(32,39)(33,34)(36,40)(37,41)(38,44)(42,43);(13,17,18,26)(14,20,15,22)(16,23,24,25)(19,27,21,28)(29,32)(30,36)(31,38)(33,40)(34,39)(35,42)(37,43)(41,44)

Polycyclic presentation:

2^9|a^2=fg,b^2=fh,c^2=h,d^2=fg,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, 4773685);