#9815242

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

4075 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)(3,9,4,8)(5,10,13,14)(11,16,12,15)(17,18,22,27)(19,23,30,24)(20,25,31,29)(21,28,32,26)(33,34,38,41)(35,42,37,39)(36,44,46,40)(43,47,45,48);(1,3,7,4)(2,8,6,9)(5,12,13,11)(10,16,14,15)(17,19)(18,23)(20,28)(21,29)(22,30)(24,27)(25,32)(26,31)(33,35,38,37)(34,39,41,42)(36,43,46,45)(40,47,44,48);(17,19,22,30)(18,24,27,23)(20,28,31,26)(21,29,32,25)(33,36,38,46)(34,40,41,44)(35,43,37,45)(39,47,42,48);(1,4,7,3)(2,9,6,8)(5,12,13,11)(10,16,14,15)(17,19)(18,23)(20,28)(21,29)(22,30)(24,27)(25,32)(26,31)(33,36)(34,40)(35,43)(37,45)(38,46)(39,47)(41,44)(42,48);(1,5,7,13)(2,10,6,14)(3,11,4,12)(8,15,9,16)(17,20)(18,25)(19,28)(21,24)(22,31)(23,32)(26,30)(27,29)(33,37,38,35)(34,39,41,42)(36,45,46,43)(40,47,44,48);(1,6,7,2)(3,9,4,8)(5,10,13,14)(11,15,12,16)(17,21,22,32)(18,26,27,28)(19,29,30,25)(20,23,31,24)(33,34)(35,39)(36,40)(37,42)(38,41)(43,47)(44,46)(45,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=h,d^2=ghi,e^2=g,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, 9722438);