#8924643

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

4234 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8831839);