#6439901

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
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	#2414	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2145 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6347097);