#8755338

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

4127 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,c^2=ghi,d^2=ghi,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, 8662534);