#6441802

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438955
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×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2214 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)(7,9)(8,10)(11,12)(13,14,20,36,19,25,18,24)(15,26,35,21,30,42,29,39)(16,31,34,40,33,22,17,23)(27,43,28,41,44,38,32,37);(1,2)(3,4)(5,7,8,11)(6,9,10,12)(13,15)(14,21)(16,32)(17,27)(18,35)(19,30)(20,29)(22,38)(23,41)(24,42)(25,39)(26,36)(28,33)(31,43)(34,44)(37,40);(2,4)(6,10)(9,12)(13,16,20,34,19,33,18,17)(14,22,36,23,25,31,24,40)(15,27,35,28,30,44,29,32)(21,37,42,43,39,41,26,38);(2,4)(13,17,18,33,19,34,20,16)(14,23,24,22,25,40,36,31)(15,28,29,27,30,32,35,44)(21,38,26,41,39,43,42,37);(16,33)(17,34)(22,31)(23,40)(27,44)(28,32)(37,41)(38,43)

Polycyclic presentation:

2^9|a^2=fi,b^2=gi,c^2=fi,d^2=f,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei,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, 6348998);