#405887

order	512
nilpotency class	3
isoclinism	isoclinic class of #393906
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2487
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1003 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)(7,18,22,24)(8,14)(9,15,28,36)(10,16)(11,30)(12,17)(13,21,32,23)(19,27,35,29)(20,26)(25,33)(31,34);(1,2)(3,4)(5,7,11,23,12,22,10,21)(6,13,16,18,17,32,30,24)(8,15,25,35,26,36,31,19)(9,20,29,33,28,14,27,34);(2,4)(5,8,10,25,12,26,11,31)(6,14,16,33,17,20,30,34)(7,19,21,15,22,35,23,36)(9,24,27,13,28,18,29,32);(5,9)(6,15)(7,20)(8,24)(10,27)(11,29)(12,28)(13,25)(14,22)(16,35)(17,36)(18,26)(19,30)(21,34)(23,33)(31,32)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 313083);