#5331665

order	512
nilpotency class	2
isoclinism	isoclinic class of #5211343
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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1356 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=gh,b^2=fg,c^2=ghi,d^2=i,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,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, 5238861);