#8927982

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=gh,e^2=gh,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, 8835178);