#8187000

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	#2404	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4133 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=g,d^2=hi,e^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8094196);