#8097192

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

4081 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=gh,d^2=g,e^2=h,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, 8004388);