#1907268

order	512
nilpotency class	2
isoclinism	isoclinic class of #1506685
automorphism	?
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

1298 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:

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

Polycyclic presentation:

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

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, 1814464);