#8810102

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

4178 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)(17,18,22,23)(19,27,28,24)(20,25,29,31)(21,26,30,32)(33,34)(35,40)(36,42)(37,38)(39,41)(43,48)(44,45)(46,47);(2,4)(5,13)(7,9)(11,16)(18,23)(21,30)(24,27)(25,31)(33,35,39,36)(34,40,41,42)(37,46,43,44)(38,47,48,45);(2,4)(3,10)(8,12)(11,16)(19,28)(20,29)(24,27)(25,31)(34,41)(35,36)(37,43)(45,47);(1,3,6,10)(2,7,4,9)(5,11,13,16)(8,14,12,15)(17,19)(18,24)(20,26)(21,25)(22,28)(23,27)(29,32)(30,31)(33,36,39,35)(34,42,41,40)(37,46,43,44)(38,47,48,45);(1,4,6,2)(3,9,10,7)(5,12,13,8)(11,15,16,14)(17,20)(18,25)(19,26)(21,24)(22,29)(23,31)(27,30)(28,32)(33,37,39,43)(34,38,41,48)(35,44,36,46)(40,45,42,47);(1,5,6,13)(2,8,4,12)(3,11,10,16)(7,14,9,15)(17,21)(18,26)(19,25)(20,24)(22,30)(23,32)(27,29)(28,31)(33,38)(34,43)(35,45)(36,47)(37,41)(39,48)(40,46)(42,44)

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=h,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8717298);