#8471179

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

4166 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=gi,d^2=i,e^2=hi,f^2=hi,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, 8378375);