#8158704

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	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3984 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8065900);