#6542289

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3561	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3021 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=i,d^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=ei,f^b=fi,g^b=gi,f^c=fi,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, 6449485);