#415116

order	512
nilpotency class	3
isoclinism	isoclinic class of #396117
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2495
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

856 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)(5,6,12,17,10,16,11,18)(7,13,27,31,20,32,21,36)(8,14,28,33,22,34,23,30)(9,24,19,15,25,35,26,29);(5,7)(6,13)(8,9)(10,20)(11,27)(12,21)(14,24)(15,30)(16,32)(17,36)(18,31)(19,23)(22,25)(26,28)(29,33)(34,35);(2,4)(5,8,10,22)(6,14)(7,19,20,26)(9,21,25,27)(11,28,12,23)(13,29)(15,32)(16,34)(17,30)(18,33)(24,36)(31,35);(5,9,12,19,10,25,11,26)(6,15,18,24,16,29,17,35)(7,8,21,23,20,22,27,28)(13,30,31,14,32,33,36,34)

Polycyclic presentation:

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

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, 322312);