#390283

order	512
nilpotency class	3
isoclinism	isoclinic class of #30396
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	#23421	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄⋊C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1198 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=fh,d^2=ef,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,e^b=ei

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, 297479);