#1978601

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, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1466 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=g,c^2=fghi,d^2=gh,e^2=hi,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, 1885797);