#8931067

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

4136 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)(4,5)(6,14)(7,9)(10,15)(11,13)(12,16)(17,18,22,23)(19,24,27,20)(21,29,30,25)(26,28,32,31)(35,41)(39,46)(42,44)(47,48);(1,3,7,4)(2,8,9,5)(6,11,15,12)(10,16,14,13)(17,19)(18,20)(21,26)(22,27)(23,24)(25,28)(29,31)(30,32)(33,34)(35,39)(36,43)(37,38)(40,45)(41,46)(42,48)(44,47);(2,9)(3,4)(6,15)(13,16)(18,23)(20,24)(21,30)(26,32)(34,38)(35,41)(40,43)(42,44);(1,4,7,3)(2,8,9,5)(6,12,15,11)(10,16,14,13)(17,20)(18,19)(21,28)(22,24)(23,27)(25,26)(29,32)(30,31)(33,35)(34,39)(36,44)(37,41)(38,46)(40,48)(42,45)(43,47);(1,5,7,8)(2,4,9,3)(6,13,15,16)(10,12,14,11)(17,19)(18,24)(20,23)(21,26)(22,27)(25,31)(28,29)(30,32)(33,36,37,45)(34,40,38,43)(35,42,41,44)(39,47,46,48);(1,6)(2,10)(3,11)(4,12)(5,13)(7,15)(8,16)(9,14)(17,21,22,30)(18,25,23,29)(19,26,27,32)(20,28,24,31)(33,34)(35,39)(36,40)(37,38)(41,46)(42,47)(43,45)(44,48)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,d^2=gh,e^2=i,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, 8838263);