#416315

order	512
nilpotency class	3
isoclinism	isoclinic class of #395174
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	#2580
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

730 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)(7,18,22,29)(8,13)(9,24)(10,15)(11,16)(12,21,28,36)(14,23)(17,26)(19,33,34,32)(20,27,35,30)(25,31);(1,2)(3,4)(5,7,11,22)(6,12)(8,19,23,34)(9,20,25,35)(10,21,26,36)(13,30)(14,27)(15,18)(16,28)(17,29)(24,33)(31,32);(2,4)(5,8,10,9,11,23,26,25)(6,13,15,24,16,14,17,31)(7,19,21,20,22,34,36,35)(12,27,18,32,28,30,29,33);(5,9,10,23,11,25,26,8)(6,14,17,24,16,13,15,31)(7,20,21,34,22,35,36,19)(12,27,29,33,28,30,18,32)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250148, #72250154, #72250174, …
maximal normal subgroup of	#72250148, #72250154, #72250174, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 323511);