#418309

order	512
nilpotency class	3
isoclinism	isoclinic class of #417601
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60456	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1661 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|c^2=i,d^2=e,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,g^c=gi,h^c=hi,f^d=fi,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, 325505);