#403115

order	512
nilpotency class	3
isoclinism	isoclinic class of #395181
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1232 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 310311);