#467804

order	512
nilpotency class	3
isoclinism	isoclinic class of #455570
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
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

860 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:

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

Polycyclic presentation:

2^9|a^2=fg,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,h^a=hi,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, 375000);