#8339958

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	#48600	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4080 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=i,d^2=gi,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8247154);