#359446

order	512
nilpotency class	3
isoclinism	isoclinic class of #196338
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3449	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3876 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)(3,4)(5,6,10,16,11,15,9,14)(7,12,20,28,19,27,18,26)(8,21,22,31,23,13,24,30)(17,25,32,35,33,29,34,36);(2,4)(5,7)(6,12)(8,17)(9,18)(10,20)(11,19)(13,29)(14,26)(15,27)(16,28)(21,25)(22,32)(23,33)(24,34)(30,36)(31,35);(5,8)(6,13)(7,17)(9,22)(10,24)(11,23)(12,25)(14,30)(15,21)(16,31)(18,32)(19,33)(20,34)(26,35)(27,29)(28,36);(2,4)(5,9,11,10)(6,14,15,16)(7,18,19,20)(8,22,23,24)(12,26,27,28)(13,30,21,31)(17,32,33,34)(25,35,29,36)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240678, #72251694, #72261199, …
maximal normal subgroup of	#72240678, #72251694, #72261199, …
maximal quotient of	#72238542, #72262014

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 266642);