#5482452

order	512
nilpotency class	2
isoclinism	isoclinic class of #5473759
automorphism	Aut(G) order 4194304
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1603 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)(17,18)(19,23)(20,24)(21,22)(25,30)(26,29)(27,28)(31,32)(33,34,37,40)(35,41,43,38)(36,39,44,46)(42,47,48,45);(1,2,4,6)(3,5,7,8)(10,13)(14,16)(18,22)(20,27)(23,29)(25,31)(36,44)(39,46)(42,48)(45,47);(1,2)(3,5)(4,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,23)(20,28)(21,22)(24,27)(25,32)(26,29)(30,31)(33,35,37,43)(34,38,40,41)(36,42,44,48)(39,45,46,47);(9,11)(10,14)(12,15)(13,16)(17,19,21,26)(18,23,22,29)(20,25,27,31)(24,30,28,32)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,46)(45,48);(1,3)(2,5)(4,7)(6,8)(9,11)(10,14)(12,15)(13,16)(17,20)(18,24)(19,25)(21,27)(22,28)(23,30)(26,31)(29,32)(33,36)(34,39)(35,42)(37,44)(38,45)(40,46)(41,47)(43,48)

Polycyclic presentation:

2^9|a^2=g,b^2=fghi,c^2=g,d^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, 5389648);