#8757113

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

3979 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)(3,9)(4,7)(5,8)(6,12)(10,14)(11,15)(13,16)(19,25)(22,26)(27,31)(30,32)(34,37)(39,42)(40,45)(46,48);(1,3,7,5)(2,8,4,9)(6,13,14,11)(10,16,12,15)(17,18)(19,26)(20,28)(21,24)(22,25)(23,29)(27,30)(31,32)(36,44)(40,45)(41,47)(46,48);(2,4)(8,9)(10,12)(15,16)(17,18,21,24)(19,22,25,26)(20,23,29,28)(27,30,31,32)(33,34,38,37)(35,39,43,42)(36,40,44,45)(41,46,47,48);(1,4,7,2)(3,9,5,8)(6,12,14,10)(11,16,13,15)(17,18)(19,22)(20,23)(21,24)(25,26)(27,30)(28,29)(31,32)(33,35,38,43)(34,39,37,42)(36,41,44,47)(40,46,45,48);(1,5,7,3)(2,9,4,8)(6,13,14,11)(10,16,12,15)(17,19,21,25)(18,22,24,26)(20,27,29,31)(23,30,28,32)(33,36)(34,40)(35,41)(37,45)(38,44)(39,46)(42,48)(43,47);(1,6)(2,10)(3,11)(4,12)(5,13)(7,14)(8,15)(9,16)(17,20)(18,23)(19,27)(21,29)(22,30)(24,28)(25,31)(26,32)(33,37,38,34)(35,42,43,39)(36,45,44,40)(41,48,47,46)

Polycyclic presentation:

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

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, 8664309);