#8456007

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=ghi,d^2=g,e^2=gi,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, 8363203);