#459753

order	512
nilpotency class	3
isoclinism	isoclinic class of #455572
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
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

921 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=efi,b^2=ef,d^2=ef,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,h^b=hi,d^c=de,f^c=fi,g^c=gi,h^c=hi,e^d=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, 366949);