#8445252

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

4073 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8352448);