#8722257

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	#2407	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3977 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8629453);