#508198

order	512
nilpotency class	4
isoclinism	isoclinic class of #508070
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1149 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 1149 conjugacy classes of subgroups — too many to draw. Only its 124 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) = 24
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415394);