#8196344

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3897 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,d^2=gh,e^2=h,f^2=ghi,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, 8103540);