#418974

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
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2303 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,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, 326170);