#8430471

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

4214 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gh,d^2=gh,e^2=g,f^2=g,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, 8337667);