#412964

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	#2420
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

968 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:

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

Polycyclic presentation:

2^9|b^2=ef,c^2=e,d^2=h,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,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, 320160);