#8127490

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	#56547	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3918 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=i,d^2=h,e^2=h,f^2=g,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^c=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, 8034686);