#6671913

order	512
nilpotency class	3
isoclinism	isoclinic class of #6668044
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#23871	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2161 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=i,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6579109);