#217886

order	512
nilpotency class	3
isoclinism	isoclinic class of #24378
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³

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	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

630 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)(3,6)(4,5)(7,8)(9,10)(11,16)(12,15)(13,14)(17,18,22,25,21,20,27,24)(19,26,29,31,28,23,30,32);(2,5)(3,7)(9,11,12,13)(10,14,15,16)(17,19,27,30,21,28,22,29)(18,23,24,31,20,26,25,32);(1,3,4,7)(2,6,5,8)(10,14)(11,13)(15,16)(17,20)(18,21)(19,23)(22,24)(25,27)(26,28)(29,32)(30,31);(2,5)(6,8)(10,15)(14,16)(18,20)(23,26)(24,25)(31,32)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 125082);