#4912022

order	512
nilpotency class	2
isoclinism	isoclinic class of #4861280
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

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

1238 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 56 points:

(2,5)(6,8)(9,10,14,11)(12,17,20,15)(13,18,22,16)(19,23,21,24)(25,26)(27,33)(28,36)(29,30)(31,34)(32,37)(35,38)(39,40)(41,42,46,44)(43,47,51,49)(45,52,53,48)(50,55,56,54);(10,11)(15,17)(16,18)(23,24)(26,30)(27,34)(32,36)(35,40)(41,43,46,51)(42,47,44,49)(45,50,53,56)(48,54,52,55);(9,11,14,10)(12,17,20,15)(13,18,22,16)(19,24,21,23)(25,27)(26,31)(28,35)(29,34)(30,33)(32,38)(36,39)(37,40)(41,43)(42,47)(44,49)(45,50)(46,51)(48,54)(52,55)(53,56);(1,2)(3,6)(4,5)(7,8)(9,12,14,20)(10,15,11,17)(13,21,22,19)(16,24,18,23)(25,26)(27,31)(28,36)(29,30)(32,37)(33,34)(35,39)(38,40)(41,44,46,42)(43,49,51,47)(45,52,53,48)(50,55,56,54);(1,3,4,7)(2,6,5,8)(9,13,14,22)(10,16,11,18)(12,19,20,21)(15,23,17,24)(25,28,29,37)(26,32,30,36)(27,35,34,40)(31,38,33,39)(41,45,46,53)(42,48,44,52)(43,50,51,56)(47,54,49,55)

Polycyclic presentation:

2^9|a^2=gh,b^2=fh,c^2=fg,d^2=gh,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef

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, 4819218);