#8932337

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) = 256

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

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=gh,d^2=gh,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8839533);