#229624

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂²×D₄	C₂²×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

727 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=h,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fh,d^b=de,f^c=fh

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 136820);