#114584

order	512
nilpotency class	3
isoclinism	isoclinic class of #114579
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1089 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)(3,7,8,5)(4,6)(9,10,14,19,15,20,13,18)(11,21,24,16,25,31,23,32)(12,17,27,34,28,36,29,35)(22,33,26,30,37,39,38,40);(1,3)(2,5)(4,8)(6,7)(9,11,15,25)(10,16,20,32)(12,26,28,38)(13,24,14,23)(17,33,36,39)(18,31,19,21)(22,29,37,27)(30,34,40,35);(1,2)(3,5)(4,6)(7,8)(9,12)(10,17)(11,22)(13,29)(14,27)(15,28)(16,30)(18,35)(19,34)(20,36)(21,33)(23,38)(24,26)(25,37)(31,39)(32,40)

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,c^b=cf,d^b=di,e^b=eh,f^b=fi,g^b=ghi,d^c=dhi,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 21780);