#8174363

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	#2675	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4003 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)(3,4,10,8)(5,13,14,9)(11,15,16,12)(18,21)(19,20)(26,29)(27,28)(33,34)(35,42)(36,39)(37,41)(38,44)(40,45)(43,47)(46,48);(2,7)(3,10)(9,13)(11,16)(17,18,23,21)(19,25,20,24)(22,26,30,29)(27,32,28,31)(33,35)(34,38)(36,43)(37,44)(39,46)(40,47)(41,42)(45,48);(2,7)(3,10)(9,13)(11,16)(17,19,23,20)(18,24,21,25)(22,27,30,28)(26,31,29,32)(33,36,37,45)(34,39,41,40)(35,43,44,48)(38,46,42,47);(1,3,6,10)(2,8,7,4)(5,11,14,16)(9,15,13,12)(17,20,23,19)(18,25,21,24)(22,28,30,27)(26,32,29,31)(33,36)(34,40)(35,43)(37,45)(38,47)(39,41)(42,46)(44,48);(1,4,6,8)(2,3,7,10)(5,12,14,15)(9,11,13,16)(17,21,23,18)(19,25,20,24)(22,29,30,26)(27,32,28,31)(33,34)(35,38)(36,39)(37,41)(40,45)(42,44)(43,46)(47,48);(1,5)(2,9)(3,11)(4,12)(6,14)(7,13)(8,15)(10,16)(17,22,23,30)(18,26,21,29)(19,27,20,28)(24,31,25,32)(33,35)(34,38)(36,43)(37,44)(39,46)(40,47)(41,42)(45,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=ghi,d^2=h,e^2=h,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8081559);