#416255

order	512
nilpotency class	3
isoclinism	isoclinic class of #395174
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2535
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

946 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=eh,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 323451);