#9913304

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

4211 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=gh,d^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9820500);