#8455984

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

4188 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=ghi,e^2=i,f^2=g,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, 8363180);