#8926312

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

4146 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:

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

Polycyclic presentation:

2^9|b^2=gi,c^2=g,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8833508);