#8930987

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

3981 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,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, 8838183);