#6671903

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	#23863	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2198 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)(13,14,18,22)(15,25,27,20)(16,29,31,38)(17,24,19,23)(21,33,36,30)(26,40,41,44)(28,35,32,37)(34,43,39,42);(1,2)(3,4)(5,6,8,9)(7,10,11,12)(13,15,18,27)(14,20,22,25)(16,26,31,41)(17,32,19,28)(21,34,36,39)(23,37,24,35)(29,44,38,40)(30,43,33,42);(2,4)(13,16)(14,21)(15,26)(17,30)(18,31)(19,33)(20,34)(22,36)(23,29)(24,38)(25,39)(27,41)(28,42)(32,43)(35,40)(37,44);(5,7,8,11)(6,10,9,12)(14,22)(15,27)(21,36)(23,24)(26,41)(28,32)(29,38)(42,43);(2,4)(6,9)(10,12)(13,17,18,19)(14,23,22,24)(15,28,27,32)(16,30,31,33)(20,35,25,37)(21,29,36,38)(26,42,41,43)(34,40,39,44)

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=fi,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, 6579099);