#4427344

order	512
nilpotency class	2
isoclinism	isoclinic class of #4417862
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

1690 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)(7,10)(8,12)(13,14,18,15)(16,19,24,21)(17,22,26,20)(23,28,25,27)(30,34)(35,41)(36,42)(43,44);(5,6,9,11)(7,8,10,12)(14,15)(19,21)(20,22)(27,28)(29,30)(31,35)(32,36)(33,34)(37,43)(38,41)(39,44)(40,42);(1,2,3,4)(5,7,9,10)(6,8,11,12)(13,15,18,14)(16,21,24,19)(17,22,26,20)(23,28,25,27)(29,30)(31,35)(32,36)(33,34)(37,43)(38,41)(39,44)(40,42);(5,8)(6,10)(7,11)(9,12)(13,16,18,24)(14,19,15,21)(17,25,26,23)(20,28,22,27)(29,31,33,38)(30,35,34,41)(32,39,40,37)(36,44,42,43);(13,17)(14,20)(15,22)(16,23)(18,26)(19,27)(21,28)(24,25)(29,32,33,40)(30,36,34,42)(31,37,38,39)(35,43,41,44)

Polycyclic presentation:

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

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, 4334540);