#5880888

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

1423 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=gh,c^2=hi,d^2=gi,e^2=fh,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, 5788084);