#8926721

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

3975 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:

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

Polycyclic presentation:

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

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, 8833917);