#392433

order	512
nilpotency class	3
isoclinism	isoclinic class of #190022
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#381
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1525 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299629);