#8269703

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

4062 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)(3,5,10,8)(4,12,13,9)(11,15,16,14)(33,34,39,43)(35,38,45,47)(36,42,46,40)(37,41,44,48);(2,7)(3,10)(4,13)(14,15)(17,18)(19,22)(20,24)(21,29)(23,26)(25,31)(27,32)(28,30)(33,35,39,45)(34,38,43,47)(36,37,46,44)(40,48,42,41);(2,7)(5,8)(9,12)(14,15)(17,19,23,20)(18,24,26,22)(21,27,31,28)(25,32,29,30)(33,36,39,46)(34,40,43,42)(35,44,45,37)(38,41,47,48);(1,3,6,10)(2,8,7,5)(4,11,13,16)(9,15,12,14)(17,20,23,19)(18,24,26,22)(21,28,31,27)(25,32,29,30)(33,37)(34,41)(35,36)(38,42)(39,44)(40,47)(43,48)(45,46);(1,4)(2,9)(3,11)(5,14)(6,13)(7,12)(8,15)(10,16)(17,21,23,31)(18,25,26,29)(19,27,20,28)(22,30,24,32)(33,38)(34,35)(36,41)(37,42)(39,47)(40,44)(43,45)(46,48);(1,5,6,8)(2,10,7,3)(4,14,13,15)(9,16,12,11)(17,22,23,24)(18,19,26,20)(21,30,31,32)(25,27,29,28)(33,36)(34,42)(35,37)(38,41)(39,46)(40,43)(44,45)(47,48)

Polycyclic presentation:

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

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, 8176899);