#9913245

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

4405 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=gh,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, 9820441);