#370193

order	512
nilpotency class	3
isoclinism	isoclinic class of #24390
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1374 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fghi,c^2=eg,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=di,f^c=fi,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 277389);