#475289

order	512
nilpotency class	3
isoclinism	isoclinic class of #465214
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

747 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=h,d^2=g,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,f^b=fi,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,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, 382485);