#497817

order	512
nilpotency class	3
isoclinism	isoclinic class of #188770
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#432
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #1770 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

461 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 24 points:

(2,6)(7,8)(9,10,14,20,13,18,22,17,15,19,12,16,23,24,11,21);(1,2,5,6)(3,7,4,8)(9,11,13,14,15,22,23,12)(10,16,24,17,19,20,18,21);(1,3,5,4)(2,7,6,8)(9,11,13,14,15,22,23,12)(10,17,18,16,19,21,24,20);(1,4,5,3)(2,7,6,8)(9,12,13,11,15,14,23,22)(10,16,18,21,19,20,24,17)

Polycyclic presentation:

2^9|a^2=g,b^2=ef,c^2=e,d^2=e,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,f^a=fi,d^b=deh,g^b=gi,d^c=di,g^c=gi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405013);