#108361

order	512
nilpotency class	3
isoclinism	isoclinic class of #108352
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²⋊#95
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

780 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

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, 15557);