#8739421

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

3797 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)(4,15)(5,9)(6,10)(7,14)(8,13)(12,16)(17,18,23,20)(19,24,21,25)(22,26,30,28)(27,31,29,32)(33,34,38,42)(35,43,46,39)(36,47,45,40)(37,41,44,48);(1,3)(2,7)(4,12)(5,13)(6,14)(8,16)(9,15)(10,11)(17,19,23,21)(18,24,20,25)(22,27,30,29)(26,31,28,32)(33,35,38,46)(34,39,42,43)(36,37,45,44)(40,41,47,48);(1,2,6,10)(3,7,14,11)(4,8,13,15)(5,9,12,16)(17,19,23,21)(18,25,20,24)(22,27,30,29)(26,32,28,31)(33,36,38,45)(34,40,42,47)(35,44,46,37)(39,48,43,41);(1,2)(3,11)(4,8)(5,16)(6,10)(7,14)(9,12)(13,15)(17,20,23,18)(19,24,21,25)(22,28,30,26)(27,31,29,32)(33,35)(34,39)(36,44)(37,45)(38,46)(40,48)(41,47)(42,43);(1,4)(2,8)(3,12)(5,14)(6,13)(7,16)(9,11)(10,15)(17,21,23,19)(18,24,20,25)(22,29,30,27)(26,31,28,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,48)(45,47);(1,5,6,12)(2,9,10,16)(3,13,14,4)(7,15,11,8)(17,22,23,30)(18,26,20,28)(19,27,21,29)(24,31,25,32)(33,37,38,44)(34,41,42,48)(35,45,46,36)(39,47,43,40)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=h,d^2=gh,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,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, 8646617);