#8446011

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	#2671	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3959 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=h,d^2=h,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8353207);