#8425426

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	#2676	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3950 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=i,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=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, 8332622);