#378902

order	512
nilpotency class	3
isoclinism	isoclinic class of #23994
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2482
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

940 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 286098);