#5482500

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

1457 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=hi,c^2=g,d^2=fg,e^2=f,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, 5389696);