#8437796

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

3946 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)(3,8,10,7)(4,9,13,12)(11,15,16,14)(18,21)(19,20)(26,29)(27,28)(33,34,39,38)(35,40,46,45)(36,42,37,41)(43,48,44,47);(3,10)(7,8)(11,16)(14,15)(17,18,23,21)(19,25,20,24)(22,26,30,29)(27,32,28,31)(33,35,39,46)(34,40,38,45)(36,43,37,44)(41,47,42,48);(1,3)(2,7)(4,11)(5,8)(6,10)(9,14)(12,15)(13,16)(17,19,23,20)(18,24,21,25)(22,27,30,28)(26,31,29,32)(33,36,39,37)(34,41,38,42)(35,43,46,44)(40,47,45,48);(1,3)(2,8)(4,11)(5,7)(6,10)(9,15)(12,14)(13,16)(17,20,23,19)(18,25,21,24)(22,28,30,27)(26,32,29,31)(33,35)(34,40)(36,43)(37,44)(38,45)(39,46)(41,47)(42,48);(1,4,6,13)(2,9,5,12)(3,11,10,16)(7,14,8,15)(17,21,23,18)(19,25,20,24)(22,29,30,26)(27,32,28,31)(33,37,39,36)(34,42,38,41)(35,44,46,43)(40,48,45,47);(1,5,6,2)(3,8,10,7)(4,12,13,9)(11,15,16,14)(17,22,23,30)(18,26,21,29)(19,27,20,28)(24,31,25,32)(33,38,39,34)(35,45,46,40)(36,42,37,41)(43,48,44,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,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, 8344992);