#8295615

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

4278 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:

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

Polycyclic presentation:

2^9|b^2=hi,d^2=i,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8202811);