#301950

order	512
nilpotency class	3
isoclinism	isoclinic class of #301949
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1240 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 1240 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)(4,6)(7,8)(9,10)(11,19)(12,18)(13,17)(14,16)(15,20)(21,24)(22,23);(1,3)(2,5)(4,8)(6,7)(9,10,14,18,13,17,12,16)(11,15,22,24,21,23,20,19);(1,2)(3,7,8,5)(4,6)(9,10,13,17)(11,15,21,23)(12,16,14,18)(19,22,24,20);(9,11,12,20,13,21,14,22)(10,15,16,19,17,23,18,24)

Polycyclic presentation:

2^9|b^2=eh,c^2=f,d^2=e,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=gi,c^b=cg,f^b=fi,g^b=gh,g^c=ghi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 209146);