#8360928

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

4074 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,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, 8268124);