#494714

order	512
nilpotency class	4
isoclinism	isoclinic class of #480476
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	#35149	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

2159 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)(7,15,19,29)(8,12)(9,14,16,23)(10,13)(11,18)(17,28,32,36)(20,27,30,34)(21,26)(22,25)(24,31)(33,35);(5,7,16,18)(6,11,23,29)(8,17,30,31)(9,22,10,19)(12,24,34,36)(13,25,14,15)(20,33,21,32)(26,35,27,28);(2,4)(5,8,10,21)(6,12,13,26)(7,17,19,32)(9,20,16,30)(11,24,25,35)(14,27,23,34)(15,28,29,36)(18,31,22,33);(6,13)(11,25)(12,26)(14,23)(15,29)(24,35)(27,34)(28,36)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=i,e^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,h^a=hi,e^b=ehi,g^b=ghi,h^b=hi,g^e=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72279131, #72279197, #72279206, …
maximal normal subgroup of	#72279131, #72279197, #72279206, …
maximal quotient of	#72317771, #72318034, #72318038

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401910);