#392929

order	512
nilpotency class	3
isoclinism	isoclinic class of #32888
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#385
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1425 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 20 points:

(2,4)(5,6,10,14,11,15,9,17)(7,16,13,12,19,20,18,8);(5,7,10,13,11,19,9,18)(6,12,14,20,15,8,17,16);(1,2,3,4);(5,8)(6,13)(7,17)(9,20)(10,16)(11,12)(14,19)(15,18)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=f,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,g^a=gi,e^b=ei,g^d=gi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 300125);