#8928016

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

3996 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)(4,12)(5,7)(6,11)(9,13)(10,14)(15,16)(17,18)(19,22)(20,29)(21,25)(23,26)(24,28)(27,32)(30,31)(33,34,38,36)(35,41,43,39)(37,40,45,44)(42,47,48,46);(1,3)(2,6)(4,10)(5,11)(7,8)(9,14)(12,15)(13,16)(17,19)(18,22)(20,21)(23,27)(24,30)(25,29)(26,32)(28,31)(33,34,38,36)(35,39,43,41)(37,44,45,40)(42,47,48,46);(2,7)(4,13)(6,8)(10,16)(19,27)(20,28)(22,32)(24,29)(33,35)(34,39)(36,41)(37,42)(38,43)(40,46)(44,47)(45,48);(1,2)(3,6)(4,9)(5,7)(8,11)(10,14)(12,13)(15,16)(17,20)(18,24)(19,21)(22,30)(23,28)(25,32)(26,29)(27,31)(33,36,38,34)(35,41,43,39)(37,44,45,40)(42,47,48,46);(1,3)(2,8)(4,10)(5,11)(6,7)(9,15)(12,14)(13,16)(17,21)(18,25)(19,28)(20,27)(22,24)(23,31)(26,30)(29,32)(33,37,38,45)(34,40,36,44)(35,42,43,48)(39,46,41,47);(1,4,5,13)(2,9,7,12)(3,10,11,16)(6,14,8,15)(17,22)(18,19)(20,30)(21,24)(23,32)(25,28)(26,27)(29,31)(33,35)(34,39)(36,41)(37,42)(38,43)(40,46)(44,47)(45,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,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, 8835212);