#499100

order	512
nilpotency class	4
isoclinism	isoclinic class of #498890
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	#37129	C₂
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×D₄ ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1339 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 1339 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)(5,6)(7,13)(8,12)(9,11)(10,14)(15,19)(16,17)(18,20);(1,2)(3,4)(5,6)(7,10)(8,16)(9,11)(12,17)(13,14)(15,18)(19,20);(1,3,4,2)(5,7)(6,10)(8,14)(9,15)(11,18)(12,13)(16,19)(17,20);(6,11)(10,18)(12,16)(13,19)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406296);