#165883

order	512
nilpotency class	3
isoclinism	isoclinic class of #165873
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#23930	C₂
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

1361 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 1361 conjugacy classes of subgroups — too many to draw. Only its 158 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:

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

Polycyclic presentation:

2^9|b^2=gh,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fi,e^b=eg,f^b=fh,e^c=eh,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257796, #72257859, #72257864
maximal normal subgroup of	#72257796, #72257859, #72257864
maximal quotient of	#72265113, #72307464

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 73079);