#6535707

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533105
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	#36121	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2557 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284321, #72284834
maximal normal subgroup of	#72284321, #72284834

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6442903);