#8193836

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	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4125 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=gh,d^2=ghi,e^2=ghi,f^2=g,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, 8101032);