#10202096

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

4059 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)(3,8)(4,6)(5,12)(7,11)(9,14)(10,15)(13,16)(17,18,22,23)(19,25,26,20)(21,29,30,24)(27,28,32,31)(33,34,38,41)(35,40,44,39)(36,37,47,45)(42,48,46,43);(1,3,6,11)(2,7,4,8)(5,10,14,13)(9,15,12,16)(17,19,22,26)(18,20,23,25)(21,27,30,32)(24,28,29,31)(33,34,38,41)(35,40,44,39)(36,45,47,37)(42,43,46,48);(2,4)(7,8)(9,12)(15,16)(18,23)(19,26)(24,29)(27,32)(33,35,38,44)(34,39,41,40)(36,42,47,46)(37,43,45,48);(1,4,6,2)(3,8,11,7)(5,12,14,9)(10,16,13,15)(17,18)(19,20)(21,24)(22,23)(25,26)(27,28)(29,30)(31,32)(33,35)(34,40)(36,46)(37,43)(38,44)(39,41)(42,47)(45,48);(1,3)(2,8)(4,7)(5,13)(6,11)(9,15)(10,14)(12,16)(17,20)(18,19)(21,28)(22,25)(23,26)(24,27)(29,32)(30,31)(33,36,38,47)(34,37,41,45)(35,42,44,46)(39,43,40,48);(1,5)(2,9)(3,10)(4,12)(6,14)(7,15)(8,16)(11,13)(17,21,22,30)(18,24,23,29)(19,27,26,32)(20,28,25,31)(33,37)(34,36)(35,43)(38,45)(39,42)(40,46)(41,47)(44,48)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gh,d^2=hi,e^2=gh,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10109292);