#8926705

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	#58961	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3987 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gi,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8833901);