#6456522

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	#35191	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2022 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

(5,6)(7,10)(8,9)(11,12)(13,14,18,22,19,15,17,21)(16,20,27,28,25,23,26,24);(1,2)(3,4)(13,15,19,14)(16,24,25,28)(17,21,18,22)(20,27,23,26);(2,4)(6,9)(10,12)(13,16,18,27,19,25,17,26)(14,20,22,28,15,23,21,24);(2,4)(13,16,17,26,19,25,18,27)(14,20,21,24,15,23,22,28);(5,7,8,11)(6,10,9,12)(16,25)(20,23)(24,28)(26,27)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=fi,d^2=f,e^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,d^c=di,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, 6363718);