#405018

order	512
nilpotency class	3
isoclinism	isoclinic class of #396611
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2398
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

685 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)(5,6,12,28,11,15,10,9)(7,16,20,13,21,18,19,30)(8,22,25,14,27,24,26,32)(17,34,35,31,36,29,23,33);(5,7,10,20,11,21,12,19)(6,13,9,18,15,30,28,16)(8,23,26,17,27,35,25,36)(14,31,22,29,32,33,24,34);(5,8,10,25,11,27,12,26)(6,14,9,24,15,32,28,22)(7,17,19,35,21,36,20,23)(13,29,16,33,30,34,18,31);(2,4)(5,9,10,15,11,28,12,6)(7,18,20,30,21,16,19,13)(8,24,25,32,27,22,26,14)(17,31,23,34,36,33,35,29)

Polycyclic presentation:

2^9|a^2=efg,b^2=f,c^2=e,d^2=g,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,e^b=ei,f^c=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, 312214);