#499107

order	512
nilpotency class	4
isoclinism	isoclinic class of #13357
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×Q₈	C₂⁴
socle	C₂²	C₂×B(4,𝔽₂)
prime radical	#52167	C₂
LCS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

1222 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 1222 conjugacy classes of subgroups — too many to draw. Only its 178 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 20 points:

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

Polycyclic presentation:

2^9|c^2=e,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,f^a=fi,g^a=ghi,c^b=cg,f^b=fh,g^b=gi,f^c=fi,g^c=gi,h^c=hi,g^f=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406303);