#108029

order	512
nilpotency class	3
isoclinism	isoclinic class of #108026
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²⋊#95
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1024 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,d^b=di,e^b=ehi,f^b=fh,d^c=dhi,e^c=ei,f^c=fhi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 15225);