#449971

order	512
nilpotency class	3
isoclinism	isoclinic class of #425091
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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

995 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=e,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,g^c=gi,h^c=hi,f^d=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, 357167);