#9713338

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	#2733	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3871 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)(3,11)(4,14)(5,8)(6,10)(7,9)(12,15)(13,16)(17,18,24,28)(19,25,31,32)(20,21,22,26)(23,30,29,27)(33,34,40,43)(35,44,37,38)(36,45,46,41)(39,47,48,42);(1,3,8,7)(2,9,5,11)(4,13,6,12)(10,16,14,15)(17,19)(18,25)(20,30)(21,29)(22,27)(23,26)(24,31)(28,32)(33,35,40,37)(34,38,43,44)(36,42,46,47)(39,45,48,41);(1,4,8,6)(2,10,5,14)(3,12,7,13)(9,15,11,16)(17,20,24,22)(18,26,28,21)(19,27,31,30)(23,25,29,32)(33,36,40,46)(34,41,43,45)(35,42,37,47)(38,39,44,48);(1,5,8,2)(3,11,7,9)(4,10,6,14)(12,15,13,16)(17,21,24,26)(18,20,28,22)(19,29,31,23)(25,30,32,27)(33,37,40,35)(34,38,43,44)(36,42,46,47)(39,41,48,45);(1,6,8,4)(2,10,5,14)(3,13,7,12)(9,15,11,16)(17,22,24,20)(18,26,28,21)(19,30,31,27)(23,25,29,32)(33,38)(34,35)(36,39)(37,43)(40,44)(41,42)(45,47)(46,48);(1,7,8,3)(2,11,5,9)(4,13,6,12)(10,16,14,15)(17,23,24,29)(18,27,28,30)(19,26,31,21)(20,25,22,32)(33,39)(34,42)(35,45)(36,44)(37,41)(38,46)(40,48)(43,47)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=h,d^2=h,e^2=i,f^2=i,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, 9620534);