#6542526

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533105
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×#3520	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3185 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)(13,14,18,26,17,21,16,20)(15,22,23,19,25,27,24,28);(5,6)(7,10)(8,9)(11,12)(13,15,17,25)(14,19,21,28)(16,23,18,24)(20,27,26,22);(2,4)(6,9)(10,12)(13,16,17,18)(14,20,21,26)(15,23,25,24)(19,27,28,22);(5,7)(6,10)(8,11)(9,12)(14,21)(19,28)(20,26)(22,27);(7,11)(10,12)(13,16,17,18)(14,20,21,26)(15,23,25,24)(19,27,28,22)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6449722);