#8750795

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	#2414	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3831 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)(3,12)(4,10)(5,13)(6,8)(7,9)(11,15)(14,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,34,38,41)(35,36,43,39)(37,45,46,40)(42,47,48,44);(1,3,7,6)(2,8,9,12)(4,13,14,11)(5,10,15,16)(18,22)(23,25)(24,29)(30,31)(33,35,38,43)(34,39,41,36)(37,42,46,48)(40,47,45,44);(2,9)(4,14)(8,12)(11,13)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,35)(34,39)(36,41)(37,42)(38,43)(40,47)(44,45)(46,48);(1,4)(2,10)(3,11)(5,8)(6,13)(7,14)(9,16)(12,15)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31)(33,36,38,39)(34,35,41,43)(37,44,46,47)(40,42,45,48);(1,5)(2,11)(3,10)(4,8)(6,16)(7,15)(9,13)(12,14)(17,20,21,28)(18,24,22,29)(19,26,27,32)(23,30,25,31)(33,37)(34,40)(35,42)(36,44)(38,46)(39,47)(41,45)(43,48);(1,6,7,3)(2,8,9,12)(4,11,14,13)(5,10,15,16)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,d^2=i,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8657991);