#8291633

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×#41	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3908 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8198829);