#8097260

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

4094 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=h,d^2=g,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, 8004456);