#6466546

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438017
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2082 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:

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

Polycyclic presentation:

2^9|b^2=fg,c^2=ghi,d^2=h,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^d=ei

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, 6373742);