#412619

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

878 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)(5,6,11,16)(7,18)(8,23,26,14)(9,27,25,33)(10,15,12,17)(13,28)(19,34)(20,30)(21,32)(22,31)(24,36)(29,35);(5,7)(6,13)(8,24)(9,20)(10,28)(11,22)(12,21)(14,30)(15,31)(16,32)(17,18)(19,25)(23,34)(26,35)(27,36)(29,33);(2,4)(5,8,10,25,11,26,12,9)(6,14,15,27,16,23,17,33)(7,19,21,24,22,20,28,35)(13,29,18,34,32,36,31,30);(5,9,12,26,11,25,10,8)(6,14,15,27,16,23,17,33)(7,20,21,35,22,19,28,24)(13,30,31,36,32,34,18,29)

Polycyclic presentation:

2^9|a^2=f,c^2=e,d^2=h,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=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, 319815);