#8994375

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

4143 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)(3,8)(4,10)(5,11)(6,14)(7,9)(12,16)(13,15)(17,18,22,23)(19,25,27,21)(20,28,29,24)(26,31,32,30)(33,34,38,41)(35,42,36,39)(37,40,46,45)(43,48,44,47);(1,3,7,4)(2,8,9,10)(5,14,15,12)(6,13,16,11)(17,19,22,27)(18,21,23,25)(20,26,29,32)(24,31,28,30)(33,35,38,36)(34,39,41,42)(37,43,46,44)(40,47,45,48);(2,9)(3,4)(5,15)(6,16)(18,23)(20,29)(21,25)(26,32)(33,34,38,41)(35,39,36,42)(37,45,46,40)(43,48,44,47);(1,4,7,3)(2,10,9,8)(5,12,15,14)(6,11,16,13)(17,20)(18,24)(19,26)(21,31)(22,29)(23,28)(25,30)(27,32)(33,34)(35,39)(36,42)(37,40)(38,41)(43,47)(44,48)(45,46);(1,5,7,15)(2,11,9,13)(3,12,4,14)(6,8,16,10)(17,21)(18,19)(20,30)(22,25)(23,27)(24,32)(26,28)(29,31)(33,36,38,35)(34,39,41,42)(37,44,46,43)(40,47,45,48);(1,6)(2,12)(3,13)(4,11)(5,8)(7,16)(9,14)(10,15)(17,18)(19,21)(20,24)(22,23)(25,27)(26,31)(28,29)(30,32)(33,37)(34,40)(35,43)(36,44)(38,46)(39,47)(41,45)(42,48)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=ghi,d^2=i,e^2=gh,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, 8901571);