#391615

order	512
nilpotency class	3
isoclinism	isoclinic class of #232181
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2380
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

684 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=e,d^2=eg,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=def,f^b=fi,h^b=hi,d^c=de,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, 298811);