#470975

order	512
nilpotency class	3
isoclinism	isoclinic class of #461251
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2644
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

667 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fh,c^2=ef,d^2=ei,e^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,g^b=gi,d^c=de,e^c=ei,f^c=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, 378171);