#8324321

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

4099 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,d^2=h,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, 8231517);