#8925303

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

3986 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,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, 8832499);