#8492771

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

4081 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)(3,10)(4,13)(5,8)(6,9)(7,15)(11,14)(12,16)(17,18,23,27)(19,24,28,32)(20,22,26,31)(21,30,29,25)(33,34)(35,41)(36,44)(37,38)(39,40)(42,46)(43,45)(47,48);(1,3)(2,5)(4,11)(6,8)(7,14)(9,10)(12,15)(13,16)(19,28)(21,29)(24,32)(25,30)(34,40)(37,41)(42,44)(45,48);(1,4)(2,7)(3,11)(5,14)(6,15)(8,12)(9,13)(10,16)(17,19,23,28)(18,24,27,32)(20,29,26,21)(22,25,31,30)(33,35,39,38)(34,41,40,37)(36,43,46,47)(42,48,44,45);(1,4)(2,7)(3,12)(5,16)(6,15)(8,11)(9,13)(10,14)(17,20)(18,22)(19,29)(21,28)(23,26)(24,25)(27,31)(30,32)(33,36)(34,42)(35,43)(37,45)(38,47)(39,46)(40,44)(41,48);(1,2)(3,10)(4,7)(5,8)(6,9)(11,16)(12,14)(13,15)(17,21,23,29)(18,25,27,30)(19,20,28,26)(22,24,31,32)(33,37)(34,35)(36,45)(38,40)(39,41)(42,43)(44,47)(46,48);(1,5,6,10)(2,8,9,3)(4,14,15,16)(7,12,13,11)(17,22)(18,26)(19,25)(20,27)(21,24)(23,31)(28,30)(29,32)(33,38,39,35)(34,37,40,41)(36,43,46,47)(42,48,44,45)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8399967);