#499016

order	512
nilpotency class	4
isoclinism	isoclinic class of #498821
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#60450	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1088 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)(7,12)(8,14)(9,22)(10,27)(11,16)(13,18)(15,25)(17,28)(19,30)(20,31)(21,35)(23,33)(24,29)(26,32)(34,36);(1,2)(3,4)(5,7)(6,12)(8,20)(9,23)(10,17)(11,19)(13,32)(14,31)(15,34)(16,30)(18,26)(21,24)(22,33)(25,36)(27,28)(29,35);(1,3,4,2)(5,7)(6,13)(8,17)(9,24)(10,20)(11,19)(12,22)(14,29)(15,31)(16,33)(18,25)(21,23)(26,36)(27,34)(28,35)(30,32);(5,8,11,10)(6,14,16,27)(7,17,19,20)(9,21,26,25)(12,28,30,31)(13,29,33,34)(15,22,35,32)(18,24,23,36)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250460, #72250469, #72250485, …
maximal normal subgroup of	#72250460, #72250469, #72250485, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406212);