#249324

order	512
nilpotency class	3
isoclinism	isoclinic class of #249319
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#35565	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1707 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 1707 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=h,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, 156520);