#8165161

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4064 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=gh,e^2=hi,f^2=h,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, 8072357);