#391521

order	512
nilpotency class	3
isoclinism	isoclinic class of #232181
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

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

762 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)(11,20,23,15)(12,17)(13,16)(14,18)(19,29)(21,37,38,31)(22,32,27,30)(24,40,39,34)(25,35)(26,36)(28,33);(1,2,4,5)(3,6,7,8)(9,11)(10,15)(12,24)(13,27)(14,23)(16,32)(17,34)(18,20)(19,30)(21,25)(22,29)(26,39)(28,38)(31,35)(33,37)(36,40);(1,3)(2,6)(4,7)(5,8)(10,16,18,19)(15,30,20,32)(17,35,36,33)(31,34,37,40);(2,5)(3,7)(9,12,14,26)(10,17,18,36)(11,21,23,38)(13,28,29,25)(15,31,20,37)(16,33,19,35)(22,39,27,24)(30,40,32,34)

Polycyclic presentation:

2^9|a^2=h,b^2=eg,c^2=g,d^2=i,f^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, 298717);