#8696265

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

4200 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)(3,5)(4,12)(6,7)(8,10)(9,14)(11,15)(13,16)(17,18,23,27)(19,24,30,31)(20,29,21,25)(22,28,32,26)(33,34,38,42)(35,40,36,39)(37,41,46,45)(43,48,44,47);(2,7)(3,10)(9,12)(11,16)(17,19)(18,24)(20,26)(21,28)(22,29)(23,30)(25,32)(27,31)(33,34)(35,39)(36,40)(37,41)(38,42)(43,47)(44,48)(45,46);(1,3)(2,8)(4,11)(5,7)(6,10)(9,15)(12,13)(14,16)(17,20,23,21)(18,25,27,29)(19,28,30,26)(22,31,32,24)(33,35,38,36)(34,39,42,40)(37,43,46,44)(41,47,45,48);(1,3)(2,8)(4,11)(5,7)(6,10)(9,15)(12,13)(14,16)(17,21,23,20)(18,25,27,29)(19,28,30,26)(22,24,32,31)(33,34)(35,39)(36,40)(37,45)(38,42)(41,46)(43,48)(44,47);(1,4,6,14)(2,9,7,12)(3,11,10,16)(5,13,8,15)(17,22,23,32)(18,26,27,28)(19,29,30,25)(20,31,21,24)(33,36,38,35)(34,40,42,39)(37,43,46,44)(41,47,45,48);(1,5,6,8)(2,3,7,10)(4,13,14,15)(9,11,12,16)(17,19)(18,24)(20,28)(21,26)(22,25)(23,30)(27,31)(29,32)(33,37,38,46)(34,41,42,45)(35,43,36,44)(39,47,40,48)

Polycyclic presentation:

2^9|a^2=gi,c^2=gi,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=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, 8603461);