#10181048

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	#50822	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3965 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)(3,8,11,15)(4,6,13,16)(5,10,12,14)(17,18)(19,28)(20,30)(21,27)(22,24)(23,25)(26,31)(29,32);(1,3,7,11)(2,8,9,15)(4,12,13,5)(6,14,16,10)(17,19,23,22)(18,24,25,28)(20,29,31,27)(21,26,32,30)(33,34,39,37)(35,42,36,40)(38,41,46,45)(43,48,44,47);(2,9)(3,11)(4,13)(10,14)(18,25)(20,31)(24,28)(27,29)(33,35,39,36)(34,40,37,42)(38,43,46,44)(41,47,45,48);(1,4)(2,6)(3,12)(5,11)(7,13)(8,14)(9,16)(10,15)(17,20)(18,26)(19,29)(21,28)(22,27)(23,31)(24,32)(25,30)(33,36,39,35)(34,40,37,42)(38,44,46,43)(41,47,45,48);(1,5,7,12)(2,10,9,14)(3,13,11,4)(6,8,16,15)(17,21,23,32)(18,27,25,29)(19,30,22,26)(20,24,31,28)(33,37,39,34)(35,42,36,40)(38,41,46,45)(43,47,44,48);(1,6)(2,4)(3,14)(5,15)(7,16)(8,12)(9,13)(10,11)(17,22,23,19)(18,28,25,24)(20,27,31,29)(21,30,32,26)(33,38)(34,41)(35,43)(36,44)(37,45)(39,46)(40,47)(42,48)

Polycyclic presentation:

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

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, 10088244);