#379826

order	512
nilpotency class	3
isoclinism	isoclinic class of #214688
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#35660	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2104 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:

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

Polycyclic presentation:

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

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, 287022);