#206760

order	512
nilpotency class	3
isoclinism	isoclinic class of #23848
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
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 ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

908 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)(3,7)(4,5)(6,8)(10,14,15,16)(17,18,21,26,22,27,20,25)(19,24,28,31,29,32,30,23);(2,5)(6,7)(9,10,11,14,13,15,12,16)(17,19)(18,23)(20,28)(21,30)(22,29)(24,25)(26,32)(27,31);(1,3,4,8)(2,6,5,7)(17,19)(18,24)(20,30)(21,28)(22,29)(23,25)(26,31)(27,32);(9,11,13,12)(10,14,15,16)(17,20,22,21)(18,25,27,26)(19,28,29,30)(23,24,31,32)

Polycyclic presentation:

2^9|a^2=g,b^2=efgh,c^2=f,d^2=h,e^2=h,g^2=i,b^a=be,c^a=cf,e^b=eh,g^b=gh,d^c=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 113956);