#470619

order	512
nilpotency class	3
isoclinism	isoclinic class of #464719
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2631
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

844 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,f^b=fi,d^c=de,e^c=ei,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, 377815);