#359510

order	512
nilpotency class	3
isoclinism	isoclinic class of #196278
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	C₂×#3449	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3922 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)(8,20,22,31)(9,14)(10,15)(11,16)(13,21,29,24)(17,32,35,36)(18,26)(19,27)(23,30)(25,34,28,33);(2,4)(5,7)(6,12)(8,17)(9,18)(10,23)(11,19)(13,28)(14,26)(15,30)(16,27)(20,32)(21,33)(22,35)(24,34)(25,29)(31,36);(5,8)(6,13)(7,17)(9,21)(10,24)(11,22)(12,25)(14,20)(15,31)(16,29)(18,33)(19,35)(23,34)(26,36)(27,28)(30,32);(2,4)(5,9,11,10)(6,14,16,15)(7,18,19,23)(8,21,22,24)(12,26,27,30)(13,20,29,31)(17,33,35,34)(25,36,28,32)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285486, #72285557, #72286235, …
maximal normal subgroup of	#72285486, #72285557, #72286235, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 266706);