#249396

order	512
nilpotency class	3
isoclinism	isoclinic class of #249391
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
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1726 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 1726 conjugacy classes of subgroups — too many to draw. Only its 152 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)(11,15)(12,16)(13,21)(14,17)(18,24)(19,22)(20,23);(3,8)(5,6)(9,11,14,13)(10,15,17,21)(12,18,20,19)(16,22,23,24);(2,5)(3,8)(6,7)(9,12)(10,16)(11,18)(13,19)(14,20)(15,22)(17,23)(21,24);(9,13,14,11)(10,15,17,21)(12,19,20,18)(16,22,23,24)

Polycyclic presentation:

2^9|b^2=h,d^2=h,f^2=i,b^a=be,c^a=cf,d^a=deh,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

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 156592);