#466489

order	512
nilpotency class	3
isoclinism	isoclinic class of #456505
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	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

765 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=efg,c^2=i,d^2=ef,e^2=i,f^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,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, 373685);