#383155

order	512
nilpotency class	3
isoclinism	isoclinic class of #213808
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

1971 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)(7,16,19,11)(8,20)(9,15)(10,14)(12,18,29,23)(13,21)(17,33,35,28)(22,32)(24,31)(25,27)(26,34,30,36);(1,2)(3,4)(5,7)(6,11)(8,17)(9,23)(10,19)(12,27)(13,30)(14,16)(15,29)(18,25)(20,28)(21,36)(22,35)(24,34)(26,31)(32,33);(2,4)(5,7)(6,12)(8,17)(9,23)(10,19)(11,15)(13,28)(14,29)(16,27)(18,25)(20,26)(21,36)(22,35)(24,34)(30,32)(31,33);(5,8,10,22)(6,13,14,31)(7,17,19,35)(9,24,25,21)(11,26,16,30)(12,28,29,33)(15,20,27,32)(18,36,23,34)

Polycyclic presentation:

2^9|a^2=h,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,h^b=hi,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, 290351);