#4469003

order	512
nilpotency class	2
isoclinism	isoclinic class of #4465744
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

1681 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)(8,10)(14,18)(15,21)(20,25)(22,27)(29,30,34,32)(31,37,38,35)(33,40,41,36)(39,42,44,43);(5,6,9,12)(7,8,11,10)(15,21)(19,24)(22,27)(26,28)(30,32)(31,38)(33,41)(42,43);(5,7,9,11)(6,10,12,8)(13,14,17,18)(15,19,21,24)(16,20,23,25)(22,26,27,28)(29,31,34,38)(30,35,32,37)(33,39,41,44)(36,42,40,43);(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,15,17,21)(14,19,18,24)(16,22,23,27)(20,26,25,28)(29,32,34,30)(31,35,38,37)(33,40,41,36)(39,42,44,43);(5,8,9,10)(6,11,12,7)(13,16,17,23)(14,20,18,25)(15,22,21,27)(19,26,24,28)(29,33)(30,36)(31,39)(32,40)(34,41)(35,42)(37,43)(38,44)

Polycyclic presentation:

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

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, 4376199);