#4862851

order	512
nilpotency class	2
isoclinism	isoclinic class of #4861280
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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

1824 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)(7,11)(10,12)(13,14)(15,18)(16,21)(17,22)(19,23)(20,26)(24,27)(25,28)(29,30,33,36)(31,34,38,41)(32,35,39,42)(37,40,43,44);(6,9)(10,12)(14,15)(16,23)(20,22)(24,25)(29,31,33,38)(30,34,36,41)(32,37,39,43)(35,40,42,44);(5,6,8,9)(7,10,11,12)(13,15,18,14)(16,21,23,19)(17,22,26,20)(24,28,25,27)(29,30)(31,34)(32,35)(33,36)(37,40)(38,41)(39,42)(43,44);(1,2)(3,4)(13,16)(14,19)(15,21)(17,25)(18,23)(20,28)(22,27)(24,26)(29,31)(30,34)(32,37)(33,38)(35,40)(36,41)(39,43)(42,44);(5,7)(6,10)(8,11)(9,12)(13,17,18,26)(14,20,15,22)(16,24,23,25)(19,27,21,28)(29,32,33,39)(30,35,36,42)(31,37,38,43)(34,40,41,44)

Polycyclic presentation:

2^9|a^2=fg,b^2=fg,c^2=h,e^2=g,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, 4770047);