#9913194

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	#55677	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4045 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gh,d^2=g,f^2=g,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, 9820390);