#449497

order	512
nilpotency class	3
isoclinism	isoclinic class of #417530
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3344 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)(7,15)(8,20,22,12)(9,23)(10,14)(11,18)(13,25)(16,30,32,29)(17,19)(21,28,36,34)(24,27)(26,31,33,35);(1,2)(3,4)(5,7,10,19)(6,11,14,27)(8,16,22,32)(9,24,25,18)(12,26,20,33)(13,17,23,15)(21,35,36,31)(28,30,34,29);(2,4)(5,8)(6,12)(7,16)(9,21)(10,22)(11,26)(13,28)(14,20)(15,29)(17,30)(18,31)(19,32)(23,34)(24,35)(25,36)(27,33);(5,6)(7,17)(8,12)(9,23)(10,14)(11,24)(13,25)(15,19)(16,30)(18,27)(20,22)(21,34)(26,35)(28,36)(29,32)(31,33)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72251164, #72255954, #72256186, …
maximal normal subgroup of	#72251164, #72255954, #72256186, …
maximal quotient of	#72267312, #72267319

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 356693);