#8097324

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

4171 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=g,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

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, 8004520);