#6584581

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573477
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	#2541	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#124 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1956 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fg,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,e^b=eh,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi

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, 6491777);