#213491

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, prime radical	C₂³	C₂×C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

576 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=f,d^2=eg,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,c^b=chi,d^c=dhi

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, 120687);