#8746020

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	#2635	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3818 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=i,d^2=gi,e^2=gh,f^2=gh,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, 8653216);