#161371

order	512
nilpotency class	3
isoclinism	isoclinic class of #7654
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×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	#2856	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

1076 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=gh,d^2=i,e^2=h,f^2=i,b^a=be,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, 68567);