#10036711

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	#39326	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4039 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)(3,8)(4,14)(5,11)(6,15)(7,9)(10,13)(12,16)(17,18)(19,27)(20,25)(21,28)(22,24)(23,30)(26,32)(29,31)(33,34,38,42)(35,43,45,39)(36,46,47,40)(37,41,44,48);(1,3,7,12)(2,8,9,16)(4,6,13,11)(5,14,15,10)(17,19,22,30)(18,23,24,27)(20,26,31,28)(21,29,32,25)(33,35,38,45)(34,39,42,43)(36,37,47,44)(40,41,46,48);(2,9)(3,12)(4,13)(5,15)(18,24)(20,31)(23,27)(26,28)(33,36,38,47)(34,40,42,46)(35,44,45,37)(39,48,43,41);(1,4,7,13)(2,10,9,14)(3,11,12,6)(5,8,15,16)(17,20)(18,25)(19,28)(21,27)(22,31)(23,32)(24,29)(26,30)(33,36)(34,40)(35,37)(38,47)(39,41)(42,46)(43,48)(44,45);(1,5,7,15)(2,11,9,6)(3,10,12,14)(4,8,13,16)(17,18)(19,27)(20,25)(21,28)(22,24)(23,30)(26,32)(29,31)(33,37)(34,41)(35,36)(38,44)(39,40)(42,48)(43,46)(45,47);(1,6,7,11)(2,5,9,15)(3,13,12,4)(8,14,16,10)(17,21)(18,26)(19,29)(20,27)(22,32)(23,31)(24,28)(25,30)(33,34)(35,39)(36,40)(37,41)(38,42)(43,45)(44,48)(46,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=ghi,d^2=gh,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9943907);