#385902

order	512
nilpotency class	3
isoclinism	isoclinic class of #213808
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

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	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1494 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)(7,11,17,29)(8,19)(9,15)(10,14)(12,18,28,24)(13,25)(16,30,20,35)(21,32)(22,31)(23,26)(27,33,36,34);(2,4)(5,7)(6,11)(8,20)(9,17)(10,24)(12,14)(13,27)(15,29)(16,21)(18,23)(19,35)(22,33)(25,34)(26,28)(30,32)(31,36);(1,2,3,4)(5,7)(6,12)(8,16)(9,17)(10,24)(11,26)(13,30)(14,29)(15,28)(18,23)(19,27)(20,21)(22,34)(25,33)(31,35)(32,36);(5,8,9,21)(6,13,15,31)(7,16,17,20)(10,25,23,22)(11,27,29,36)(12,30,28,35)(14,32,26,19)(18,34,24,33)

Polycyclic presentation:

2^9|a^2=h,c^2=egi,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, 293098);