#448913

order	512
nilpotency class	3
isoclinism	isoclinic class of #421159
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

705 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 356109);