#8756204

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	#45991	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3938 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,d^2=h,e^2=ghi,f^2=h,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, 8663400);