#6439845

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	C₂×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2143 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)(13,14,18,23)(15,24,19,20)(16,27,29,38)(17,22,30,36)(21,32,35,25)(26,37,31,33)(28,40,41,44)(34,42,43,39);(1,2)(3,4)(6,9)(10,12)(13,15,18,19)(14,20,23,24)(16,25,29,32)(17,26,30,31)(21,27,35,38)(22,33,36,37)(28,39,41,42)(34,40,43,44);(2,4)(13,16,18,29)(14,21,23,35)(15,25,19,32)(17,28,30,41)(20,27,24,38)(22,34,36,43)(26,39,31,42)(33,40,37,44);(2,4)(13,17)(14,22)(15,26)(16,28)(18,30)(19,31)(20,33)(21,34)(23,36)(24,37)(25,39)(27,40)(29,41)(32,42)(35,43)(38,44);(5,7,8,11)(6,10,9,12)(17,30)(22,36)(26,31)(28,41)(33,37)(34,43)(39,42)(40,44)

Polycyclic presentation:

2^9|a^2=i,b^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,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, 6347041);