#249337

order	512
nilpotency class	3
isoclinism	isoclinic class of #249319
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#35659	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2016 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 2016 conjugacy classes of subgroups — too many to draw. Only its 156 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fhi,e^c=eh,f^c=fi,f^d=fh

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, 156533);