#415120

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

Polycyclic presentation:

2^9|a^2=fh,c^2=e,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,e^b=ei,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, 322316);