#8824428

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

4152 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=gi,d^2=hi,e^2=i,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8731624);