#8621895

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

4192 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=g,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,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, 8529091);