#6535614

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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	#39355	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2405 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)(14,18)(15,21)(17,24)(19,27)(22,28)(23,25);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,14,16,18)(15,22,23,19)(17,20,24,26)(21,27,25,28);(6,9)(10,12)(13,15,16,23)(14,19,18,22)(17,25,24,21)(20,27,26,28);(5,7)(6,10)(8,11)(9,12)(13,14)(15,19)(16,18)(17,20)(21,28)(22,23)(24,26)(25,27);(7,11)(10,12)(14,18)(19,22)(20,26)(27,28)

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,d^b=di,e^b=ei,g^c=gi,e^d=eh

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6442810);