#1960682

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

1362 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=fi,d^2=gh,e^2=fhi,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, 1867878);