#9863070

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

3943 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)(3,11)(4,7)(5,9)(6,15)(8,13)(10,16)(12,14)(17,18,23,21)(19,27,28,24)(20,29,30,25)(22,26,32,31)(33,34,39,36)(35,37,40,42)(38,41,47,44)(43,45,48,46);(1,3,7,13)(2,8,4,11)(5,6,12,16)(9,10,14,15)(17,19,23,28)(18,24,21,27)(20,26,30,31)(22,25,32,29)(35,40)(37,42)(43,48)(45,46);(2,4)(8,11)(9,14)(10,15)(17,18,23,21)(19,24,28,27)(20,25,30,29)(22,26,32,31)(33,35)(34,37)(36,42)(38,43)(39,40)(41,45)(44,46)(47,48);(1,4,7,2)(3,11,13,8)(5,14,12,9)(6,15,16,10)(17,20,23,30)(18,25,21,29)(19,26,28,31)(22,24,32,27)(33,36,39,34)(35,42,40,37)(38,44,47,41)(43,46,48,45);(1,5)(2,9)(3,6)(4,14)(7,12)(8,10)(11,15)(13,16)(17,21,23,18)(19,24,28,27)(20,29,30,25)(22,26,32,31)(33,37,39,42)(34,40,36,35)(38,46,47,45)(41,43,44,48);(1,6)(2,10)(3,12)(4,15)(5,13)(7,16)(8,14)(9,11)(17,22,23,32)(18,26,21,31)(19,25,28,29)(20,24,30,27)(33,38)(34,41)(35,43)(36,44)(37,45)(39,47)(40,48)(42,46)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9770266);