#8188567

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3905 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)(3,5)(4,12)(6,7)(8,10)(9,14)(11,15)(13,16)(17,18,23,25)(19,24,21,31)(20,28,30,22)(26,27,29,32)(34,36)(35,41)(40,44)(43,45);(2,7)(3,10)(9,12)(11,16)(17,19,23,21)(18,24,25,31)(20,29,30,26)(22,27,28,32)(33,34,38,36)(35,39,41,42)(37,40,46,44)(43,47,45,48);(1,3)(2,8)(4,11)(5,7)(6,10)(9,15)(12,13)(14,16)(17,20,23,30)(18,22,25,28)(19,26,21,29)(24,32,31,27)(33,35)(34,39)(36,42)(37,45)(38,41)(40,48)(43,46)(44,47);(1,3)(2,8)(4,11)(5,7)(6,10)(9,15)(12,13)(14,16)(33,34,38,36)(35,39,41,42)(37,40,46,44)(43,47,45,48);(1,4,6,14)(2,9,7,12)(3,11,10,16)(5,13,8,15)(17,21,23,19)(18,24,25,31)(20,29,30,26)(22,32,28,27)(33,36,38,34)(35,42,41,39)(37,44,46,40)(43,48,45,47);(1,5,6,8)(2,3,7,10)(4,13,14,15)(9,11,12,16)(17,22)(18,20)(19,27)(21,32)(23,28)(24,29)(25,30)(26,31)(33,37,38,46)(34,40,36,44)(35,43,41,45)(39,47,42,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=gh,d^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8095763);