#160796

order	512
nilpotency class	3
isoclinism	isoclinic class of #160780
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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	C₂³×#41	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

2068 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)(10,14,15,16)(17,18,19,22,20,21,23,24);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,16)(13,15);(18,21)(22,24);(2,6)(5,7)(9,11,13,12)(10,14,15,16)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72279308, #72338860, #72338866
maximal normal subgroup of	#72279308, #72338860, #72338866
maximal quotient of	#72279441, #72279465

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 67992);