#8324234

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

4056 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=g,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8231430);