#9717515

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

4023 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)(4,7)(6,12)(8,10)(9,14)(11,15)(13,16)(17,18)(19,20)(21,25)(22,24)(23,27)(26,28)(29,31)(30,32)(33,34)(35,40)(36,39)(37,41)(38,46)(42,48)(43,44)(45,47);(1,3)(2,8)(4,5)(6,11)(7,10)(9,15)(12,13)(14,16)(17,19)(18,23)(20,24)(21,29)(22,27)(25,28)(26,30)(31,32)(33,35)(34,40)(36,44)(37,46)(38,41)(39,43)(42,47)(45,48);(2,4)(3,10)(9,12)(11,16)(18,24)(20,23)(25,32)(28,31)(34,36)(35,43)(41,45)(42,46);(1,4,7,2)(3,5,10,8)(6,12,14,9)(11,13,16,15)(17,20,22,23)(18,19,24,27)(21,28,30,31)(25,26,32,29)(33,36,39,34)(35,44,43,40)(37,45,47,41)(38,46,48,42);(1,5)(2,3)(4,10)(6,13)(7,8)(9,11)(12,16)(14,15)(17,21)(18,25)(19,26)(20,28)(22,30)(23,31)(24,32)(27,29)(33,37,39,47)(34,41,36,45)(35,42,43,46)(38,40,48,44);(1,6,7,14)(2,9,4,12)(3,11,10,16)(5,13,8,15)(17,19)(18,23)(20,24)(21,29)(22,27)(25,28)(26,30)(31,32)(33,38)(34,42)(35,41)(36,46)(37,44)(39,48)(40,47)(43,45)

Polycyclic presentation:

2^9|d^2=h,e^2=gh,f^2=ghi,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, 9624711);