#6671840

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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	#24425	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2159 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)(13,14,18,22)(15,23,24,36)(16,20,27,34)(17,29,31,21)(19,30,32,26)(25,37,38,44)(28,41,40,35)(33,42,43,39);(1,2)(3,4)(5,6,8,9)(7,10,11,12)(15,24)(19,32)(23,36)(25,38)(26,30)(33,43)(37,44)(39,42);(2,4)(13,15,18,24)(14,19,22,32)(16,25,27,38)(17,30,31,26)(20,33,34,43)(21,36,29,23)(28,42,40,39)(35,44,41,37);(5,7,8,11)(6,10,9,12)(13,16)(14,20)(15,25)(17,28)(18,27)(19,33)(21,35)(22,34)(23,37)(24,38)(26,39)(29,41)(30,42)(31,40)(32,43)(36,44);(2,4)(6,9)(10,12)(16,27)(20,34)(25,38)(28,40)(33,43)(35,41)(37,44)(39,42)

Polycyclic presentation:

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

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, 6579036);