#9938212

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

4307 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,d^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dh,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, 9845408);