#412620

order	512
nilpotency class	3
isoclinism	isoclinic class of #396118
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

868 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)(8,24,26,14)(9,28,27,32)(10,15,12,17)(13,22)(19,30)(20,36)(21,31)(23,33)(25,29)(34,35);(5,7)(6,13)(8,25)(9,20)(10,22)(11,23)(12,21)(14,30)(15,33)(16,31)(17,18)(19,27)(24,36)(26,35)(28,34)(29,32);(2,4)(5,8,10,9,11,26,12,27)(6,14,15,32,16,24,17,28)(7,19,21,35,23,20,22,25)(13,29,18,30,31,34,33,36);(5,9,10,26,11,27,12,8)(6,14,17,28,16,24,15,32)(7,20,22,35,23,19,21,25)(13,30,18,34,31,36,33,29)

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,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, 319816);