#161407

order	512
nilpotency class	3
isoclinism	isoclinic class of #7654
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

884 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

(2,6,7,8)(9,10,13,15,14,11,12,16)(17,18)(19,25,27,22)(20,28)(21,24)(23,29)(26,30,32,31);(1,2)(3,6)(4,8)(5,7)(9,11,14,10)(12,15,13,16)(17,19,21,27)(18,22,24,25)(20,26,29,32)(23,30,28,31);(9,12,14,13)(10,15,11,16)(17,20,21,29)(18,23,24,28)(19,26,27,32)(22,30,25,31);(1,3,5,4)(2,6,7,8)(9,12,14,13)(10,16,11,15)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 68603);