#6598538

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574024
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	#38025	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1912 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fhi,c^2=f,d^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,e^d=eh

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, 6505734);