#10407770

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	#2676	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3893 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=h,d^2=ghi,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

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, 10314966);