#481506

order	512
nilpotency class	4
isoclinism	isoclinic class of #481477
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄⋊C₄	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈ ◁ C₂×#520 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1085 conjugacy classes of subgroups; only the 274 normal subgroups are shown.

normal subgroups only

This group has 1085 conjugacy classes of subgroups — too many to draw. Only its 274 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=gi,c^2=i,d^2=i,e^2=hi,f^2=i,b^a=be,e^a=eg,g^a=gi,e^b=eh,f^b=fghi,d^c=di,f^e=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 388702);