#8096978

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

3967 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=g,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

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, 8004174);