#8196357

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3975 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)(19,20,27,21)(22,30,31,24)(26,29,32,28)(34,39)(35,36)(42,46)(43,44);(2,7)(3,10)(9,12)(11,16)(17,19)(18,20)(21,25)(22,26)(23,27)(24,28)(29,30)(31,32)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,48)(45,46);(1,3)(2,8)(4,11)(5,7)(6,10)(9,15)(12,13)(14,16)(17,20,23,21)(18,19,25,27)(22,28,31,29)(24,26,30,32)(33,35,38,36)(34,40,39,41)(37,44,45,43)(42,48,46,47);(1,3)(2,8)(4,11)(5,7)(6,10)(9,15)(12,13)(14,16)(17,21,23,20)(18,19,25,27)(22,29,31,28)(24,26,30,32)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,48)(45,46);(1,4,6,14)(2,9,7,12)(3,11,10,16)(5,13,8,15)(17,22,23,31)(18,24,25,30)(19,26,27,32)(20,28,21,29)(33,36,38,35)(34,41,39,40)(37,44,45,43)(42,48,46,47);(1,5,6,8)(2,3,7,10)(4,13,14,15)(9,11,12,16)(17,19)(18,20)(21,25)(22,26)(23,27)(24,28)(29,30)(31,32)(33,37)(34,42)(35,43)(36,44)(38,45)(39,46)(40,47)(41,48)

Polycyclic presentation:

2^9|a^2=gh,c^2=gi,d^2=gh,e^2=h,f^2=ghi,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, 8103553);