#6444717

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438997
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1974 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6351913);