#413932

order	512
nilpotency class	3
isoclinism	isoclinic class of #411585
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23400	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1415 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|d^2=gh,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,e^b=ei,f^b=fi,e^c=ei

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, 321128);