#9717602

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
prime radical	#2688	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3816 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)(3,10)(4,8)(5,13)(6,7)(9,15)(11,12)(14,16)(17,18)(19,29)(20,21)(22,26)(23,31)(24,28)(25,32)(27,30)(33,34)(35,40)(36,39)(37,41)(38,43)(42,46)(44,47)(45,48);(1,2)(3,4)(5,9)(6,7)(8,10)(11,12)(13,15)(14,16)(17,19)(18,25)(20,30)(21,31)(22,23)(24,32)(26,27)(28,29)(33,35)(34,38)(36,42)(37,43)(39,45)(40,41)(44,48)(46,47);(2,7)(3,8)(9,13)(11,16)(17,20,24,22)(18,26,28,21)(19,30,32,23)(25,27,29,31)(33,36,37,44)(34,39,41,47)(35,42,43,48)(38,45,40,46);(1,3,6,8)(2,4,7,10)(5,11,15,16)(9,12,13,14)(17,21,24,26)(18,20,28,22)(19,31,32,27)(23,25,30,29);(1,4)(2,8)(3,7)(5,14)(6,10)(9,11)(12,15)(13,16)(17,22,24,20)(18,26,28,21)(19,30,32,23)(25,31,29,27)(33,35)(34,40)(36,42)(37,43)(38,41)(39,46)(44,48)(45,47);(1,5)(2,9)(3,11)(4,12)(6,15)(7,13)(8,16)(10,14)(17,23,24,30)(18,27,28,31)(19,22,32,20)(21,25,26,29)(33,34)(35,38)(36,39)(37,41)(40,43)(42,45)(44,47)(46,48)

Polycyclic presentation:

2^9|c^2=g,d^2=h,e^2=hi,f^2=hi,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, 9624798);