#9945593

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

4093 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gh,d^2=ghi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,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, 9852789);