#4379697

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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1766 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fh,c^2=fh,d^2=fhi,e^2=hi,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, 4286893);