#8932771

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

4130 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8839967);