#8097234

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

4192 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:

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

Polycyclic presentation:

2^9|a^2=ghi,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, 8004430);