#8093259

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3920 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=hi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi

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, 8000455);