#265265

order	512
nilpotency class	3
isoclinism	isoclinic class of #265257
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#62245	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1113 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 1113 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) = 128

A minimal faithful permutation representation — 4 generators on 24 points:

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

Polycyclic presentation:

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

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, 172461);