#5902016

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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1398 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=ghi,c^2=gi,d^2=fghi,e^2=fg,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, 5809212);