#8093223

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

4126 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:

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

Polycyclic presentation:

2^9|c^2=gi,d^2=g,e^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, 8000419);