#8093203

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4024 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)(6,13)(10,15)(14,16)(17,18,23,26)(19,27,30,21)(20,29,32,24)(22,28,31,25)(36,43)(40,45)(42,47)(46,48);(1,2,5,8)(3,6,9,13)(4,11,12,7)(10,16,15,14)(17,19,23,30)(18,21,26,27)(20,25,32,28)(22,29,31,24)(34,38)(35,41)(40,45)(42,47);(1,3)(2,6)(4,10)(5,9)(7,14)(8,13)(11,16)(12,15)(17,20,23,32)(18,24,26,29)(19,28,30,25)(21,31,27,22)(33,34,37,38)(35,39,41,44)(36,40,43,45)(42,46,47,48);(1,4,5,12)(2,7,8,11)(3,10,9,15)(6,14,13,16)(17,19)(18,21)(20,28)(22,29)(23,30)(24,31)(25,32)(26,27);(1,2)(3,6)(4,7)(5,8)(9,13)(10,14)(11,12)(15,16)(17,21)(18,19)(20,31)(22,32)(23,27)(24,28)(25,29)(26,30)(33,35)(34,39)(36,42)(37,41)(38,44)(40,46)(43,47)(45,48);(1,3)(2,6)(4,10)(5,9)(7,14)(8,13)(11,16)(12,15)(17,22,23,31)(18,25,26,28)(19,29,30,24)(20,21,32,27)(33,36)(34,40)(35,42)(37,43)(38,45)(39,46)(41,47)(44,48)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,d^2=gh,f^2=g,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, 8000399);