#6466137

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437764
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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2282 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)(13,14,16,20,17,19,25,24)(15,18,23,26,22,21,27,28);(2,3)(6,9)(10,11)(13,14)(15,21)(16,24)(17,19)(18,22)(20,25)(23,26)(27,28);(1,2,4,3)(5,7,8,12)(6,10,9,11)(13,15,17,22)(14,18,19,21)(16,23,25,27)(20,26,24,28);(1,3,4,2)(14,19)(18,21)(20,24)(26,28);(15,22)(18,21)(23,27)(26,28)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6373333);