#8919217

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

4214 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:

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

Polycyclic presentation:

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

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, 8826413);