#4342864

order	512
nilpotency class	2
isoclinism	isoclinic class of #4311730
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

1764 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)(5,6,10,9)(7,12,11,8)(13,14)(15,19)(16,24)(17,18)(20,26)(21,23)(22,28)(25,27)(30,32)(35,39)(36,40)(42,43);(14,17)(15,21)(20,24)(22,25)(29,30,34,32)(31,35,38,39)(33,40,41,36)(37,43,44,42);(5,7)(6,8)(9,12)(10,11)(13,15,18,21)(14,19,17,23)(16,25,26,22)(20,28,24,27)(29,31)(30,35)(32,39)(33,37)(34,38)(36,42)(40,43)(41,44);(1,2,3,4)(5,8,10,12)(6,11,9,7)(13,16)(14,20)(15,22)(17,24)(18,26)(19,27)(21,25)(23,28)(29,32,34,30)(31,35,38,39)(33,40,41,36)(37,42,44,43);(5,9,10,6)(7,12,11,8)(13,17,18,14)(15,23,21,19)(16,24,26,20)(22,28,25,27)(29,33,34,41)(30,36,32,40)(31,37,38,44)(35,42,39,43)

Polycyclic presentation:

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

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, 4250060);