#8898648

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

4143 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=i,d^2=g,e^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, 8805844);