#4416315

order	512
nilpotency class	2
isoclinism	isoclinic class of #4414082
automorphism	Aut(G) order 4194304
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

1802 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=fi,c^2=f,d^2=fh,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^c=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, 4323511);