#8270496

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	#53053	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3993 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)(3,8,10,4)(5,13,15,9)(11,12,14,16)(33,34)(35,42)(36,37)(38,39)(40,45)(41,46)(43,44)(47,48);(1,3)(2,8)(4,6)(5,14)(7,10)(9,12)(11,15)(13,16)(17,18,23,22)(19,26,20,24)(21,29,30,25)(27,31,28,32)(34,39)(36,46)(40,42)(43,48);(2,6)(3,10)(9,13)(11,14)(17,19,23,20)(18,24,22,26)(21,27,30,28)(25,31,29,32)(33,35,38,45)(34,40,39,42)(36,43,46,48)(37,47,41,44);(1,4)(2,3)(5,12)(6,10)(7,8)(9,11)(13,14)(15,16)(17,20,23,19)(18,24,22,26)(21,28,30,27)(25,31,29,32)(33,36)(34,41)(35,43)(37,39)(38,46)(40,44)(42,47)(45,48);(1,5,7,15)(2,9,6,13)(3,11,10,14)(4,12,8,16)(17,21,23,30)(18,25,22,29)(19,27,20,28)(24,31,26,32)(33,36)(34,41)(35,43)(37,39)(38,46)(40,44)(42,47)(45,48);(1,6,7,2)(3,4,10,8)(5,13,15,9)(11,12,14,16)(17,22,23,18)(19,24,20,26)(21,29,30,25)(27,31,28,32)(33,37,38,41)(34,36,39,46)(35,44,45,47)(40,48,42,43)

Polycyclic presentation:

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

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, 8177692);