#359475

order	512
nilpotency class	3
isoclinism	isoclinic class of #196122
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#575	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3868 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)(7,12)(8,20,23,13)(9,14)(10,16)(11,15)(17,33,35,26)(18,27)(19,28)(21,36,34,29)(22,32,25,30)(24,31);(2,4)(5,7,11,19)(6,12,15,28)(8,21,23,34)(9,18,10,24)(13,29,20,36)(14,27,16,31)(17,25,35,22)(26,32,33,30);(5,8)(6,13)(7,17)(9,22)(10,25)(11,23)(12,26)(14,30)(15,20)(16,32)(18,34)(19,35)(21,24)(27,36)(28,33)(29,31);(2,4)(5,9,11,10)(6,14,15,16)(7,18,19,24)(8,22,23,25)(12,27,28,31)(13,30,20,32)(17,34,35,21)(26,36,33,29)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72255827, #72256257, #72256268, …
maximal normal subgroup of	#72255827, #72256257, #72256268, …
maximal quotient of	#72303244

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 266671);