#213607

order	512
nilpotency class	3
isoclinism	isoclinic class of #162550
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1258 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 1258 conjugacy classes of subgroups — too many to draw. Only its 234 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,f^c=fi,g^c=ghi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 120803);