#9912908

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

3948 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)(3,11)(4,9)(5,8)(6,15)(7,10)(12,13)(14,16)(17,18,23,21)(19,27,29,24)(20,30,31,25)(22,32,28,26)(33,34,39,41)(35,42,36,37)(38,40,47,48)(43,46,44,45);(1,3,7,5)(2,8,10,11)(4,12,15,14)(6,13,9,16)(17,19,23,29)(18,24,21,27)(20,28,31,22)(25,26,30,32)(33,35,39,36)(34,37,41,42)(38,43,47,44)(40,45,48,46);(1,4,7,15)(2,6,10,9)(3,12,5,14)(8,13,11,16)(17,20,23,31)(18,25,21,30)(19,22,29,28)(24,32,27,26)(33,36,39,35)(34,37,41,42)(38,44,47,43)(40,45,48,46);(1,4)(2,9)(3,12)(5,14)(6,10)(7,15)(8,16)(11,13)(17,20)(18,25)(19,28)(21,30)(22,29)(23,31)(24,26)(27,32);(1,5,7,3)(2,8,10,11)(4,14,15,12)(6,13,9,16)(17,21,23,18)(19,24,29,27)(20,30,31,25)(22,32,28,26)(33,37)(34,35)(36,41)(38,46)(39,42)(40,44)(43,48)(45,47);(1,6,7,9)(2,4,10,15)(3,13,5,16)(8,12,11,14)(17,22,23,28)(18,26,21,32)(19,20,29,31)(24,30,27,25)(33,38)(34,40)(35,43)(36,44)(37,45)(39,47)(41,48)(42,46)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=g,e^2=h,f^2=h,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, 9820104);