#315563

order	512
nilpotency class	3
isoclinism	isoclinic class of #298603
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2856	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

901 conjugacy classes of subgroups; only the 250 normal subgroups are shown.

normal subgroups only

This group has 901 conjugacy classes of subgroups — too many to draw. Only its 250 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) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226826
maximal normal subgroup of	#72226826
maximal quotient of	#72224930

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 222759);