#10464573

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

4345 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,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, 10371769);