#471154

order	512
nilpotency class	3
isoclinism	isoclinic class of #460861
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
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

768 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:

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

Polycyclic presentation:

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

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, 378350);