#8193926

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
prime radical	#36500	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4052 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8101122);