#8893719

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

4223 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)(3,11,13,8)(4,9,14,12)(5,15,16,10)(18,23)(19,24)(25,30)(26,31)(33,34,38,41)(35,39,42,46)(36,37,40,45)(43,44,47,48);(2,7)(4,14)(8,11)(10,15)(17,18,22,23)(19,20,24,28)(21,25,29,30)(26,27,31,32)(33,35)(34,39)(36,43)(37,44)(38,42)(40,47)(41,46)(45,48);(2,7)(5,16)(8,11)(9,12)(21,29)(25,30)(26,31)(27,32)(35,42)(39,46)(43,47)(44,48);(1,3,6,13)(2,8,7,11)(4,15,14,10)(5,12,16,9)(17,19)(18,20)(21,26)(22,24)(23,28)(25,27)(29,31)(30,32)(33,36)(34,37)(35,43)(38,40)(39,44)(41,45)(42,47)(46,48);(1,4)(2,9)(3,10)(5,11)(6,14)(7,12)(8,16)(13,15)(17,20)(18,24)(19,23)(21,27)(22,28)(25,31)(26,30)(29,32)(33,37)(34,40)(35,44)(36,41)(38,45)(39,47)(42,48)(43,46);(1,5,6,16)(2,10,7,15)(3,12,13,9)(4,11,14,8)(17,21,22,29)(18,25,23,30)(19,26,24,31)(20,27,28,32)(33,35)(34,39)(36,43)(37,44)(38,42)(40,47)(41,46)(45,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,d^2=h,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, 8800915);