#359644

order	512
nilpotency class	3
isoclinism	isoclinic class of #188252
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3489	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3846 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313084, #72313137, #72313145
maximal normal subgroup of	#72313084, #72313137, #72313145

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 266840);