#4374494

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

1749 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)(6,9)(10,12)(13,14)(15,21)(16,20)(17,18)(19,23)(22,28)(24,27)(25,26)(29,30)(31,37)(32,40)(33,36)(34,39)(35,41)(38,42)(43,44);(14,18)(19,21)(20,26)(27,28)(29,30,33,36)(31,34,39,37)(32,40,41,35)(38,43,44,42);(1,2,3,4)(13,15)(14,19)(16,24)(17,23)(18,21)(20,28)(22,25)(26,27)(29,30)(31,37)(32,35)(33,36)(34,39)(38,43)(40,41)(42,44);(5,6)(7,10)(8,9)(11,12)(13,16,17,25)(14,20,18,26)(15,22,23,24)(19,27,21,28)(29,31)(30,34)(32,38)(33,39)(35,42)(36,37)(40,43)(41,44);(5,7,8,11)(6,10,9,12)(13,14)(15,19)(16,20)(17,18)(21,23)(22,27)(24,28)(25,26)(29,32)(30,35)(31,38)(33,41)(34,42)(36,40)(37,43)(39,44)

Polycyclic presentation:

2^9|b^2=h,c^2=fgh,d^2=fh,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, 4281690);