#8159055

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3863 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=i,d^2=g,e^2=h,f^2=h,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, 8066251);