#412316

order	512
nilpotency class	3
isoclinism	isoclinic class of #396118
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2537
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1238 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)(5,6,12,18)(7,19,22,33)(8,23,26,32)(9,27,24,35)(10,17,11,16)(13,29,34,21)(14,30,36,25)(15,20,31,28);(2,4)(5,7)(6,13)(8,24)(9,26)(10,29)(11,21)(12,22)(14,35)(15,23)(16,33)(17,19)(18,34)(20,25)(27,36)(28,30)(31,32);(5,8)(6,14)(7,20)(9,21)(10,30)(11,25)(12,26)(13,31)(15,34)(16,32)(17,23)(18,36)(19,35)(22,28)(24,29)(27,33);(5,9,11,28,12,24,10,20)(6,15,17,27,18,31,16,35)(7,8,21,25,22,26,29,30)(13,32,19,14,34,23,33,36)

Polycyclic presentation:

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

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, 319512);