#8417042

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

3975 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)(3,7,9,11)(5,8,13,12)(10,14,15,16)(17,18,24,27)(19,28,23,25)(20,22,21,26)(29,31,30,32)(34,40)(35,37)(42,47)(43,45);(2,4)(3,9)(8,12)(10,15)(17,19,24,23)(18,25,27,28)(20,30,21,29)(22,31,26,32)(33,34,39,40)(35,41,37,36)(38,42,46,47)(43,48,45,44);(5,13)(8,12)(10,15)(14,16)(17,20,24,21)(18,26,27,22)(19,29,23,30)(25,31,28,32)(33,35,39,37)(34,41,40,36)(38,45,46,43)(42,44,47,48);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,16)(13,15)(17,21,24,20)(18,22,27,26)(19,30,23,29)(25,32,28,31)(33,36)(34,35)(37,40)(38,44)(39,41)(42,43)(45,47)(46,48);(1,4,6,2)(3,7,9,11)(5,12,13,8)(10,14,15,16)(17,22)(18,20)(19,31)(21,27)(23,32)(24,26)(25,30)(28,29)(33,37,39,35)(34,41,40,36)(38,45,46,43)(42,48,47,44);(1,5,6,13)(2,8,4,12)(3,10,9,15)(7,14,11,16)(17,23,24,19)(18,25,27,28)(20,30,21,29)(22,32,26,31)(33,38,39,46)(34,42,40,47)(35,43,37,45)(36,44,41,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=h,d^2=i,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8324238);