#8446002

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	#2661	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3948 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=h,d^2=gh,e^2=i,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, 8353198);