#393279

order	512
nilpotency class	3
isoclinism	isoclinic class of #32932
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1186 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)(3,4)(5,6,8,7)(9,10)(11,15)(12,16)(13,14)(19,25)(20,28)(22,29)(24,30)(26,31)(27,32);(5,7,8,6)(10,14)(15,16)(17,18)(19,26)(20,24)(21,23)(22,27)(25,31)(28,30)(29,32);(9,11,13,12)(10,15,14,16)(17,19,20,25,21,27,28,32)(18,22,24,29,23,26,30,31);(2,4)(9,12,13,11)(10,16,14,15)(18,23)(22,26)(24,30)(29,31)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=ef,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,c^b=ci,d^b=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, 300475);