#416436

order	512
nilpotency class	3
isoclinism	isoclinic class of #396117
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

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

924 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)(7,19)(8,14)(9,27)(10,18)(11,17)(12,16)(13,21)(15,22)(20,29)(23,36)(24,32)(25,31)(26,33)(28,34)(30,35);(1,2,4,3)(5,7,10,21,11,22,12,9)(6,13,16,19,17,27,18,15)(8,23,25,35,26,34,24,20)(14,29,32,28,33,30,31,36);(1,3,4,2)(5,8,10,24,11,26,12,25)(6,14,16,31,17,33,18,32)(7,20,21,34,22,35,9,23)(13,28,19,29,27,36,15,30);(5,9,12,22,11,21,10,7)(6,15,18,27,17,19,16,13)(8,23,25,35,26,34,24,20)(14,30,32,36,33,29,31,28)

Polycyclic presentation:

2^9|b^2=fh,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, 323632);