#8755141

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

4119 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=ghi,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8662337);