#5572535

order	512
nilpotency class	2
isoclinism	isoclinic class of #5473759
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

1484 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=h,d^2=h,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=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, 5479731);