#8492773

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

4089 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)(3,10)(4,13)(5,7)(6,9)(8,14)(11,15)(12,16)(17,18,23,27)(19,24,28,32)(20,22,26,31)(21,30,29,25)(34,39)(36,44)(40,42)(43,45);(1,3,6,5)(2,7,9,10)(4,11,14,12)(8,15,13,16)(19,28)(21,29)(24,32)(25,30)(33,34,38,39)(35,40,37,42)(36,41,44,46)(43,47,45,48);(1,4)(2,8)(3,11)(5,12)(6,14)(7,15)(9,13)(10,16)(17,19,23,28)(18,24,27,32)(20,29,26,21)(22,25,31,30)(33,35,38,37)(34,40,39,42)(36,43,44,45)(41,47,46,48);(1,4)(2,8)(3,12)(5,11)(6,14)(7,16)(9,13)(10,15)(17,20)(18,22)(19,29)(21,28)(23,26)(24,25)(27,31)(30,32)(33,36)(34,41)(35,43)(37,45)(38,44)(39,46)(40,47)(42,48);(1,2)(3,10)(4,8)(5,7)(6,9)(11,16)(12,15)(13,14)(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)(44,46)(45,48);(1,5,6,3)(2,7,9,10)(4,12,14,11)(8,15,13,16)(17,22)(18,26)(19,25)(20,27)(21,24)(23,31)(28,30)(29,32)(33,37,38,35)(34,42,39,40)(36,43,44,45)(41,47,46,48)

Polycyclic presentation:

2^9|a^2=i,b^2=g,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, 8399969);