#9685567

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	#2828	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3805 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=i,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,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, 9592763);