#107256

order	512
nilpotency class	3
isoclinism	isoclinic class of #8479
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#379
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂²	C₂²⋊#95
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1006 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)(9,10,13,18,15,20,14,19)(11,21,23,30,24,34,28,16)(12,17,26,32,27,33,25,31)(22,36,39,29,38,35,37,40);(1,3)(2,4)(5,7)(6,8)(9,11,14,23,15,24,13,28)(10,16,19,21,20,30,18,34)(12,22,25,39,27,38,26,37)(17,29,31,35,33,40,32,36);(1,4)(2,3)(5,8)(6,7)(9,12)(10,17)(11,22)(13,26)(14,25)(15,27)(16,29)(18,32)(19,31)(20,33)(21,35)(23,39)(24,38)(28,37)(30,40)(34,36)

Polycyclic presentation:

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

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, 14452);