#218534

order	512
nilpotency class	3
isoclinism	isoclinic class of #32706
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	C₂²×D₈	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

622 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 125730);