#161129

order	512
nilpotency class	3
isoclinism	isoclinic class of #161128
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

1558 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,e^2=h,f^2=i,b^a=be,c^a=ci,d^a=dh,e^a=eg,e^b=eh,f^b=fgh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72309162, #72309357, #72309361
maximal normal subgroup of	#72309162, #72309357, #72309361

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 68325);