#401871

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

1234 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)(10,15)(12,24)(13,16)(17,26)(18,33)(19,28)(20,34)(21,22)(25,29)(31,32)(35,36)(37,40);(1,3)(2,5)(4,7)(6,8)(9,10,16,28,14,19,13,15)(11,17,22,31,23,32,21,26)(12,18,29,38,25,33,24,27)(20,30,34,40,36,39,35,37);(3,7)(5,8)(9,11)(10,17)(12,20)(13,21)(14,23)(15,26)(16,22)(18,30)(19,32)(24,35)(25,36)(27,37)(28,31)(29,34)(33,39)(38,40);(9,12,13,24,14,25,16,29)(10,18,15,27,19,33,28,38)(11,20,21,35,23,36,22,34)(17,30,26,37,32,39,31,40)

Polycyclic presentation:

2^9|a^2=g,b^2=ei,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,h^c=hi,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, 309067);