#416283

order	512
nilpotency class	3
isoclinism	isoclinic class of #400051
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
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

924 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=eh,d^2=h,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,f^b=fi,e^c=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, 323479);