#8108733

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	#46289	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4044 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=ghi,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi

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, 8015929);