#119599

order	512
nilpotency class	3
isoclinism	isoclinic class of #10637
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#397
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1440
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

665 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2,4,7,5,3,8,6)(9,10)(11,20)(12,24,28,38)(13,29)(14,18)(15,19)(16,22)(17,27,35,26)(21,31,25,34)(23,32)(30,33)(36,37,40,39);(9,11)(10,16)(12,25)(13,30)(14,22)(15,23)(17,34)(18,32)(19,33)(20,29)(21,28)(24,36)(26,37)(27,39)(31,35)(38,40);(1,3)(2,5)(4,6)(7,8)(9,12,13,26,15,28,14,27)(10,17,18,24,19,35,29,38)(11,21,22,37,23,25,30,39)(16,31,20,36,33,34,32,40)

Polycyclic presentation:

2^9|a^2=g,c^2=d,d^2=h,e^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,c^b=cf,d^b=dh,e^c=ei,f^c=fh,g^c=gh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 26795);