#379818

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	#35676	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2144 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:

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

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,c^b=cg,h^b=hi,d^c=de,f^c=fi,h^c=hi,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, 287014);