#4193461

order	512
nilpotency class	2
isoclinism	isoclinic class of #4131229
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

1380 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=gh,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4100657);