#8360942

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	#55456	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3915 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)(3,4,11,8)(6,9,14,13)(10,12,16,15)(17,18,23,20)(19,24,26,22)(21,25,31,28)(27,32,29,30)(34,36)(39,42)(40,44)(46,48);(1,3,7,11)(2,8,5,4)(6,10,14,16)(9,15,13,12)(17,19,23,26)(18,24,20,22)(21,29,31,27)(25,30,28,32)(35,41)(39,42)(43,47)(46,48);(6,14)(9,13)(10,16)(12,15)(19,26)(22,24)(27,29)(30,32)(33,34,38,36)(35,39,41,42)(37,40,45,44)(43,46,47,48);(1,4)(2,3)(5,11)(6,12)(7,8)(9,10)(13,16)(14,15)(17,20,23,18)(19,22,26,24)(21,25,31,28)(27,32,29,30)(33,35,38,41)(34,39,36,42)(37,43,45,47)(40,46,44,48);(1,5,7,2)(3,8,11,4)(6,13,14,9)(10,15,16,12)(17,21)(18,25)(19,27)(20,28)(22,30)(23,31)(24,32)(26,29)(33,36,38,34)(35,42,41,39)(37,44,45,40)(43,48,47,46);(1,6)(2,9)(3,10)(4,12)(5,13)(7,14)(8,15)(11,16)(17,22)(18,19)(20,26)(21,30)(23,24)(25,27)(28,29)(31,32)(33,37,38,45)(34,40,36,44)(35,43,41,47)(39,46,42,48)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8268138);