#8492772

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

4093 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:

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

Polycyclic presentation:

2^9|a^2=gi,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, 8399968);