#419748

order	512
nilpotency class	3
isoclinism	isoclinic class of #417704
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60425	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1584 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 326944);