#418975

order	512
nilpotency class	3
isoclinism	isoclinic class of #417859
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60366	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2231 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,c^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,h^c=hi,e^d=ei,f^d=fi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 326171);