#8360932

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	#54833	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4078 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=g,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, 8268128);