#8458807

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

4176 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)(17,18,23,28)(19,29,31,24)(20,21,26,25)(22,27,32,30)(33,34)(35,43)(36,40)(37,45)(38,41)(39,42)(44,47)(46,48);(1,2)(3,7)(4,8)(5,9)(6,10)(11,14)(12,15)(13,16)(17,19,23,31)(18,24,28,29)(20,22,26,32)(21,30,25,27)(33,35)(34,37)(36,44)(38,40)(39,45)(41,46)(42,43)(47,48);(1,3)(2,5)(4,11)(6,9)(7,10)(8,13)(12,16)(14,15)(17,20,23,26)(18,25,28,21)(19,22,31,32)(24,27,29,30)(33,36,39,46)(34,40,42,48)(35,44,45,41)(37,38,43,47);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,16)(13,15)(17,18)(19,29)(20,25)(21,26)(22,30)(23,28)(24,31)(27,32)(33,37)(34,35)(36,38)(39,43)(40,44)(41,48)(42,45)(46,47);(1,4,6,12)(2,8,10,15)(3,11,9,16)(5,13,7,14)(17,21,23,25)(18,26,28,20)(19,30,31,27)(22,24,32,29)(33,35)(34,37)(36,44)(38,40)(39,45)(41,46)(42,43)(47,48);(1,5,6,7)(2,9,10,3)(4,13,12,14)(8,16,15,11)(17,22)(18,27)(19,26)(20,31)(21,24)(23,32)(25,29)(28,30)(33,38)(34,41)(35,40)(36,43)(37,46)(39,47)(42,44)(45,48)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,e^2=hi,f^2=i,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, 8366003);