#9909995

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

4020 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,d^2=g,e^2=g,f^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, 9817191);