#379505

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	#2487
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

906 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=eg,d^2=eh,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,f^b=fi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,f^d=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, 286701);