#107955

order	512
nilpotency class	3
isoclinism	isoclinic class of #107946
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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1072 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)(9,10,14,19,15,20,13,18)(11,21,25,31,23,16,24,32)(12,26,28,17,29,33,27,34)(22,36,39,35,37,40,38,30);(1,3)(2,4)(5,7)(6,8)(9,11)(10,16,20,21)(12,22)(13,24)(14,25)(15,23)(17,30,34,35)(18,31,19,32)(26,40,33,36)(27,38)(28,39)(29,37);(1,4)(2,3)(5,8)(6,7)(9,12,15,29)(10,17,20,34)(11,22,23,37)(13,28,14,27)(16,30,21,35)(18,33,19,26)(24,39,25,38)(31,36,32,40)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,d^b=di,e^b=eh,f^b=fh,d^c=dh,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, 15151);