#8908090

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

4243 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)(3,9,10,4)(5,13,14,8)(11,12,16,15)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,32)(29,30)(33,34)(35,43)(36,38)(37,40)(39,42)(41,46)(44,48)(45,47);(2,7)(4,9)(8,13)(12,15)(17,19,23,20)(18,24,22,25)(21,28,30,27)(26,32,29,31)(33,35,39,37)(34,40,42,43)(36,44,46,47)(38,45,41,48);(2,7)(3,10)(5,14)(12,15)(18,22)(19,20)(26,29)(27,28)(36,46)(38,41)(44,47)(45,48);(1,3,6,10)(2,4,7,9)(5,11,14,16)(8,12,13,15)(17,20,23,19)(18,25,22,24)(21,27,30,28)(26,31,29,32)(33,36)(34,41)(35,44)(37,47)(38,42)(39,46)(40,48)(43,45);(1,4)(2,3)(5,12)(6,9)(7,10)(8,11)(13,16)(14,15)(17,21,23,30)(18,26,22,29)(19,27,20,28)(24,31,25,32)(33,37,39,35)(34,40,42,43)(36,44,46,47)(38,48,41,45);(1,5)(2,8)(3,11)(4,12)(6,14)(7,13)(9,15)(10,16)(17,22,23,18)(19,25,20,24)(21,29,30,26)(27,32,28,31)(33,38,39,41)(34,36,42,46)(35,45,37,48)(40,44,43,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,d^2=h,e^2=gh,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, 8815286);