#8097219

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

4217 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)(4,12)(5,7)(6,9)(8,14)(11,15)(13,16)(17,18,23,22)(19,24,29,28)(20,30,21,25)(26,32,27,31)(33,34)(35,36)(37,45)(38,40)(39,42)(41,48)(43,47)(44,46);(1,3,6,5)(2,7,9,10)(4,11,14,13)(8,15,12,16)(17,19,23,29)(18,24,22,28)(20,26,21,27)(25,31,30,32)(33,35)(34,38)(36,42)(37,46)(39,40)(41,44)(43,48)(45,47);(17,20,23,21)(18,25,22,30)(19,26,29,27)(24,31,28,32)(33,36,39,38)(34,40,42,35)(37,44,48,47)(41,46,45,43);(1,4)(2,8)(3,11)(5,13)(6,14)(7,15)(9,12)(10,16)(17,19)(18,24)(20,26)(21,27)(22,28)(23,29)(25,31)(30,32)(33,37)(34,41)(35,43)(36,44)(38,47)(39,48)(40,46)(42,45);(1,5,6,3)(2,7,9,10)(4,13,14,11)(8,15,12,16)(17,21,23,20)(18,25,22,30)(19,27,29,26)(24,31,28,32)(33,35)(34,36)(37,43)(38,42)(39,40)(41,44)(45,47)(46,48);(1,2)(3,7)(4,8)(5,10)(6,9)(11,15)(12,14)(13,16)(17,22,23,18)(19,28,29,24)(20,30,21,25)(26,32,27,31)(33,38,39,36)(34,40,42,35)(37,47,48,44)(41,46,45,43)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=h,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

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, 8004415);