#415119

order	512
nilpotency class	3
isoclinism	isoclinic class of #396836
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2420
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

858 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=fh,c^2=e,d^2=hi,e^2=i,f^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^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, 322315);