#9794277

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

4095 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)(3,10)(4,14)(5,12)(6,8)(7,13)(9,15)(11,16)(17,18,22,23)(19,25,26,20)(21,29,30,24)(27,28,32,31)(33,34,40,37)(35,38,41,36)(39,45,47,42)(43,44,48,46);(1,3)(2,7)(4,12)(5,9)(6,13)(8,10)(11,15)(14,16)(17,19,22,26)(18,20,23,25)(21,27,30,32)(24,28,29,31)(33,35)(34,38)(36,37)(39,43)(40,41)(42,46)(44,45)(47,48);(2,8)(4,15)(7,10)(11,12)(18,23)(19,26)(24,29)(27,32)(33,36,40,38)(34,41,37,35)(39,44,47,46)(42,48,45,43);(1,4)(2,9)(3,12)(5,7)(6,15)(8,14)(10,16)(11,13)(17,18)(19,20)(21,24)(22,23)(25,26)(27,28)(29,30)(31,32)(33,37,40,34)(35,36,41,38)(39,45,47,42)(43,44,48,46);(1,3)(2,10)(4,12)(5,14)(6,13)(7,8)(9,16)(11,15)(17,20)(18,19)(21,28)(22,25)(23,26)(24,27)(29,32)(30,31)(33,38,40,36)(34,41,37,35)(39,44,47,46)(42,43,45,48);(1,5,6,16)(2,11,8,12)(3,9,13,14)(4,7,15,10)(17,21,22,30)(18,24,23,29)(19,27,26,32)(20,28,25,31)(33,39,40,47)(34,42,37,45)(35,43,41,48)(36,44,38,46)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=gh,d^2=gh,e^2=gh,f^2=h,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, 9701473);