#8464166

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

4075 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=i,d^2=h,e^2=gi,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8371362);