#4469648

order	512
nilpotency class	2
isoclinism	isoclinic class of #4465744
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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

1793 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)(7,11)(13,14)(15,21)(16,20)(17,18)(19,23)(22,27)(24,25)(26,28)(30,34)(31,37)(32,40)(42,44);(7,11)(10,12)(14,18)(16,24)(19,21)(22,28)(29,30,33,34)(31,38,37,35)(32,41,40,36)(39,42,43,44);(5,6,8,9)(7,10,11,12)(13,15,17,23)(14,19,18,21)(16,22,24,28)(20,26,25,27)(29,30)(31,38)(32,36)(33,34)(35,37)(39,44)(40,41)(42,43);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,14)(15,21)(16,20)(17,18)(19,23)(22,27)(24,25)(26,28)(29,31,33,37)(30,35,34,38)(32,39,40,43)(36,42,41,44);(13,16,17,24)(14,20,18,25)(15,22,23,28)(19,26,21,27)(29,32,33,40)(30,36,34,41)(31,39,37,43)(35,42,38,44)

Polycyclic presentation:

2^9|b^2=h,c^2=gh,d^2=h,e^2=f,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, 4376844);