#470666

order	512
nilpotency class	3
isoclinism	isoclinic class of #454974
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2482
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

927 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=h,d^2=e,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^c=de,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, 377862);