#385516

order	512
nilpotency class	3
isoclinism	isoclinic class of #30412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

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	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1416 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=ef,d^2=efi,f^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 292712);