#4404765

order	512
nilpotency class	2
isoclinism	isoclinic class of #4379698
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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1674 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fh,c^2=gh,d^2=fg,e^2=fgi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^c=ef

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, 4311961);