#8157090

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	#40661	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4218 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=i,d^2=g,e^2=gh,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, 8064286);