#9923675

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

4364 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)(3,8)(4,9)(5,10)(6,7)(11,13)(12,16)(14,15)(17,18)(19,22)(20,23)(21,26)(24,29)(25,28)(27,31)(30,32)(33,34)(35,43)(36,40)(37,46)(38,42)(39,45)(41,47)(44,48);(2,7)(4,8)(10,15)(12,13)(17,18,22,19)(20,25,28,23)(21,24,29,26)(27,31,32,30)(33,35,38,45)(34,39,42,43)(36,37,44,47)(40,41,48,46);(18,19)(23,25)(24,26)(30,31)(33,34,38,42)(35,39,45,43)(36,40,44,48)(37,41,47,46);(1,3,6,9)(2,8,7,4)(5,11,14,16)(10,13,15,12)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30)(33,34)(35,39)(36,40)(37,41)(38,42)(43,45)(44,48)(46,47);(1,4,6,8)(2,9,7,3)(5,13,14,12)(10,11,15,16)(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,31)(29,32)(33,36)(34,40)(35,37)(38,44)(39,41)(42,48)(43,46)(45,47);(1,5)(2,10)(3,11)(4,12)(6,14)(7,15)(8,13)(9,16)(17,21)(18,24)(19,26)(20,27)(22,29)(23,30)(25,31)(28,32)(33,37)(34,41)(35,44)(36,45)(38,47)(39,48)(40,43)(42,46)

Polycyclic presentation:

2^9|b^2=h,c^2=hi,d^2=ghi,e^2=gh,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, 9830871);