#416318

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

728 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=eh,c^2=e,d^2=h,e^2=i,f^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,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320592, #72320716, #72320778, …
maximal normal subgroup of	#72320592, #72320716, #72320778, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 323514);