#5806098

order	512
nilpotency class	2
isoclinism	isoclinic class of #5605735
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, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1893 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=fgi,c^2=gi,d^2=fgh,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,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, 5713294);