#466492

order	512
nilpotency class	3
isoclinism	isoclinic class of #456514
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

812 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=efg,d^2=ef,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,e^d=ei,f^d=fi

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, 373688);