#8701386

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3957 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)(3,4)(5,9)(6,7)(8,10)(11,12)(13,14)(15,16)(17,18)(19,23)(20,28)(21,25)(22,26)(24,30)(27,31)(29,32);(2,7)(3,10)(9,16)(11,13)(17,18)(19,23)(20,24)(21,25)(22,26)(27,29)(28,30)(31,32)(33,34,38,37)(35,41,43,39)(36,44,45,40)(42,46,48,47);(17,19)(18,21)(20,27)(22,25)(23,26)(24,31)(28,29)(30,32)(33,35)(34,39)(36,42)(37,41)(38,43)(40,46)(44,47)(45,48);(1,3)(2,8)(4,7)(5,13)(6,10)(9,12)(11,15)(14,16)(17,19)(18,23)(20,27)(21,26)(22,25)(24,29)(28,31)(30,32)(33,36,38,45)(34,40,37,44)(35,42,43,48)(39,46,41,47);(1,4,6,8)(2,3,7,10)(5,14,15,12)(9,13,16,11)(17,20)(18,24)(19,27)(21,31)(22,30)(23,29)(25,32)(26,28)(33,35)(34,41)(36,42)(37,39)(38,43)(40,47)(44,46)(45,48);(1,5,6,15)(2,9,7,16)(3,11,10,13)(4,12,8,14)(17,21,22,23)(18,25,26,19)(20,29,30,31)(24,27,28,32)(33,37,38,34)(35,41,43,39)(36,40,45,44)(42,46,48,47)

Polycyclic presentation:

2^9|b^2=hi,d^2=hi,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=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, 8608582);