#9726394

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

Polycyclic presentation:

2^9|b^2=h,c^2=hi,d^2=gh,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,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, 9633590);