#416314

order	512
nilpotency class	3
isoclinism	isoclinic class of #395173
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

744 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)(7,18,22,33)(8,14)(9,26)(10,16)(11,27)(12,17)(13,21,29,23)(15,25)(19,34,36,32)(20,30,35,28)(24,31);(1,2)(3,4)(5,7)(6,13,17,29)(8,19)(9,20)(10,21)(11,23)(12,22)(14,30,15,28)(16,18,27,33)(24,35)(25,36)(26,32,31,34);(2,4)(5,8,10,24,12,25,11,9)(6,14,16,31,17,15,27,26)(7,19,21,35,22,36,23,20)(13,28,18,32,29,30,33,34);(5,9,11,25,12,24,10,8)(6,15,16,26,17,14,27,31)(7,20,23,36,22,35,21,19)(13,28,18,32,29,30,33,34)

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 323510);