#6439717

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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2202 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,d^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=dh,e^b=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, 6346913);