#6578868

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2388	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1967 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fi,d^2=fi,e^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=eh,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6486064);