#471000

order	512
nilpotency class	3
isoclinism	isoclinic class of #460131
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2760
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

931 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)(3,6)(4,5)(7,8)(9,10)(11,15,21,29)(12,16)(13,25,26,17)(14,18)(19,27,22,30)(20,35)(23,31,38,33)(24,32)(28,37)(34,39)(36,40);(9,11)(10,15,18,29)(12,22,24,19)(13,20)(14,21)(16,30)(17,28,25,35)(23,36,38,34)(26,37)(27,32)(31,40)(33,39);(2,5)(6,8)(9,12)(10,16)(11,19)(13,23)(14,24)(15,27)(17,33)(18,32)(20,36)(21,22)(25,31)(26,38)(28,39)(29,30)(34,37)(35,40);(1,3,4,7)(2,6,5,8)(9,13,14,26)(10,17,18,25)(11,20,21,37)(12,23,24,38)(15,28,29,35)(16,31,32,33)(19,34,22,36)(27,39,30,40)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=fi,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,e^b=ei,g^b=gi,d^c=de,h^c=hi,g^d=gi

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, 378196);