#114649

order	512
nilpotency class	3
isoclinism	isoclinic class of #114647
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	C₂³×#41	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1730 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)(11,21,25,33)(12,26,29,17)(13,18)(14,20)(15,19)(16,23,32,24)(22,37)(27,31)(28,36,30,35)(34,38)(39,40);(1,3,4,8)(2,5,6,7)(9,11)(10,16)(12,27)(13,24)(14,23)(15,25)(17,34)(18,33)(19,32)(20,21)(22,28)(26,37)(29,39)(30,38)(31,35)(36,40);(9,12)(10,17)(11,22)(13,28)(14,30)(15,29)(16,31)(18,35)(19,26)(20,36)(21,34)(23,27)(24,39)(25,38)(32,40)(33,37)

Polycyclic presentation:

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

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, 21845);