#8893635

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

4081 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=g,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8800831);