#9913079

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

4211 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9820275);