#391523

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

773 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)(9,10,14,16,13,19,27,18)(11,15,22,31,20,32,28,29)(12,17,26,33,24,35,25,36)(21,30,23,34,37,40,38,39);(1,2,4,5)(3,6,7,8)(9,11)(10,15)(12,23)(13,20)(14,28)(16,29)(17,34)(18,31)(19,32)(21,26)(22,27)(24,38)(25,37)(30,33)(35,39)(36,40);(1,3)(2,6)(4,7)(5,8)(10,16,19,18)(11,20)(12,24)(15,29,32,31)(17,33,35,36)(22,28)(25,26)(30,39,40,34);(2,5)(3,7)(9,12)(10,17)(11,21)(13,24)(14,26)(15,30)(16,33)(18,36)(19,35)(20,37)(22,23)(25,27)(28,38)(29,39)(31,34)(32,40)

Polycyclic presentation:

2^9|a^2=h,b^2=eg,c^2=gi,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, 298719);