#6370340

order	512
nilpotency class	2
isoclinism	isoclinic class of #534
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄²	C₂⁴
commutator	C₂²	C₂³×C₄²
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂⁴×C₄
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2514 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,e^2=f,g^2=i,b^a=bf,c^a=ci,d^a=df,c^b=cf

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6277536);