#118591

order	512
nilpotency class	3
isoclinism	isoclinic class of #10128
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#392
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

626 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 36 points:

(1,2,3,4)(6,10)(7,14)(8,11)(13,18)(15,16)(17,19)(21,22)(23,29)(24,27)(25,31)(26,28)(30,32)(33,36)(34,35);(5,6,11,18,9,13,8,10)(7,12,14,19,16,20,15,17)(21,23,29,22,26,31,25,28)(24,30,34,36,33,35,32,27);(5,7,9,16)(6,12,13,20)(8,15,11,14)(10,17,18,19)(21,24,26,33)(22,27,28,36)(23,30,31,35)(25,34,29,32)

Polycyclic presentation:

2^9|a^2=g,b^2=dh,c^2=h,d^2=h,e^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fi,c^b=cf,f^b=fi,d^c=di,e^c=ehi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 25787);