#10320396

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	#46526	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3971 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=ghi,d^2=ghi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dgi,e^b=eghi,d^c=dghi,e^d=egi,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10227592);