#8617796

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
prime radical	#2631	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4012 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8524992);