#8987195

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4196 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=gh,d^2=hi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8894391);