#8508943

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

4032 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)(3,11,13,8)(4,14,15,9)(5,10,16,12)(17,18)(19,28)(20,26)(21,25)(22,24)(23,27)(29,31)(30,32)(33,34)(35,40)(36,37)(38,42)(39,43)(41,47)(44,48)(45,46);(2,7)(4,15)(8,11)(10,12)(17,19,23,22)(18,24,27,28)(20,29,21,30)(25,31,26,32)(33,35,39,45)(34,40,43,46)(36,44,47,42)(37,48,41,38);(17,20,23,21)(18,25,27,26)(19,29,22,30)(24,31,28,32)(33,36)(34,41)(35,42)(37,43)(38,46)(39,47)(40,48)(44,45);(1,3,6,13)(2,8,7,11)(4,12,15,10)(5,14,16,9)(17,21,23,20)(18,26,27,25)(19,29,22,30)(24,31,28,32)(33,37,39,41)(34,36,43,47)(35,38,45,48)(40,42,46,44);(1,4,6,15)(2,9,7,14)(3,12,13,10)(5,11,16,8)(17,18)(19,28)(20,25)(21,26)(22,24)(23,27)(29,32)(30,31)(33,38)(34,42)(35,41)(36,46)(37,45)(39,48)(40,47)(43,44);(1,5)(2,10)(3,9)(4,11)(6,16)(7,12)(8,15)(13,14)(17,22,23,19)(18,24,27,28)(20,30,21,29)(25,31,26,32)(33,36)(34,37)(35,44)(38,46)(39,47)(40,48)(41,43)(42,45)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=hi,d^2=gi,e^2=ghi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8416139);