#219804

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

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	C₂²×C₂²⋊C₄
prime radical	C₂×#548	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1274 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fh,c^b=ch,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, 127000);