#8437777

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	#2456	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3955 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)(3,6)(4,9)(5,8)(7,12)(10,13)(11,15)(14,16)(17,18,22,27)(19,23,28,24)(20,25,29,30)(21,26,31,32)(33,34,39,38)(35,42,37,40)(36,41,44,46)(43,48,45,47);(2,3)(7,9)(8,10)(14,15)(17,19,22,28)(18,23,27,24)(20,26,29,32)(21,25,31,30)(36,44)(41,46)(43,45)(47,48);(1,2,6,3)(4,7,12,9)(5,8,13,10)(11,14,16,15)(17,19)(18,24)(20,26)(21,25)(22,28)(23,27)(29,32)(30,31)(33,35,39,37)(34,40,38,42)(36,43,44,45)(41,47,46,48);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,19)(18,23)(20,26)(21,30)(22,28)(24,27)(25,31)(29,32)(33,36)(34,41)(35,43)(37,45)(38,46)(39,44)(40,47)(42,48);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,20,22,29)(18,25,27,30)(19,26,28,32)(21,23,31,24)(33,37,39,35)(34,42,38,40)(36,45,44,43)(41,48,46,47);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,21,22,31)(18,26,27,32)(19,25,28,30)(20,23,29,24)(33,38,39,34)(35,42,37,40)(36,46,44,41)(43,48,45,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=ghi,d^2=g,e^2=h,f^2=h,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, 8344973);