#404308

order	512
nilpotency class	3
isoclinism	isoclinic class of #393982
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#62855	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2445 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:

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

Polycyclic presentation:

2^9|a^2=ef,d^2=g,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,e^c=ei,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, 311504);