#8755118

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	#2678	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3839 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=hi,d^2=i,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8662314);