#8746953

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	#40140	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4041 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=g,d^2=gh,e^2=g,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, 8654149);