#377544

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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1278 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,d^2=fhi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dei,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, 284740);