#178416

order	512
nilpotency class	3
isoclinism	isoclinic class of #178241
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

1518 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 1518 conjugacy classes of subgroups — too many to draw. Only its 140 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)(10,14)(11,12)(15,16)(18,22)(19,20)(23,24);(1,2)(3,5)(4,6)(7,8)(9,10)(11,14)(12,16)(13,15);(1,3)(2,5)(4,7)(6,8)(11,12)(14,16)(17,18)(19,22)(20,24)(21,23);(2,6)(5,8)(9,11,13,12)(10,14,15,16)(17,19,21,20)(18,22,23,24)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72279338
maximal normal subgroup of	#72279338
maximal quotient of	#72279350, #72279360

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 85612);