#114727

order	512
nilpotency class	3
isoclinism	isoclinic class of #114722
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1018 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 21923);