#8095749

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

4078 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8002945);