#217822

order	512
nilpotency class	3
isoclinism	isoclinic class of #30396
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

727 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2)(3,7)(4,6)(5,8)(10,13)(11,15)(14,16)(17,18,22,25,23,20,21,30)(19,24,29,31,27,26,28,32);(1,3)(2,5)(4,8)(6,7)(17,19,22,29,23,27,21,28)(18,24,25,31,20,26,30,32);(3,8)(5,7)(9,10,12,14)(11,16,15,13)(17,20,23,18)(19,26,27,24)(21,25,22,30)(28,31,29,32);(2,6)(5,7)(18,20)(19,27)(25,30)(28,29)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dei,f^a=fh,c^b=ce,d^b=di,d^c=di,f^c=fh

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, 125018);