#477754

order	512
nilpotency class	3
isoclinism	isoclinic class of #476106
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

744 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 384950);