#214257

order	512
nilpotency class	3
isoclinism	isoclinic class of #30396
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

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	C₂²×C₂²⋊C₄
prime radical	#34976	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1140 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)(9,10,13,16,14,12,11,15)(17,18)(19,28)(20,27)(21,26)(22,25)(23,30)(24,29)(31,32);(1,3)(2,5)(4,8)(6,7)(9,11,14,13)(10,15,12,16)(17,19,21,30)(18,23,26,28)(20,31,22,29)(24,25,32,27);(3,8)(5,7)(9,12)(10,14)(11,16)(13,15)(17,19)(18,24)(20,29)(21,30)(22,31)(23,27)(25,28)(26,32);(2,6)(5,7)(9,11,14,13)(10,15,12,16)(17,20,21,22)(18,25,26,27)(19,29,30,31)(23,24,28,32)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,d^b=dhi,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, 121453);