#6671803

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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	#23850	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2210 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6578999);