#511778

order	512
nilpotency class	4
isoclinism	isoclinic class of #511775
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

2464 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)(8,12)(9,13)(10,14)(11,18)(16,27)(17,28)(19,24)(20,25)(21,26)(22,30)(23,31)(29,35)(32,34)(33,36);(5,7)(6,11)(8,17)(9,18)(10,16)(12,23)(13,15)(14,22)(19,31)(20,29)(21,30)(24,28)(25,33)(26,27)(32,36)(34,35);(7,16)(9,21)(11,22)(15,27)(17,29)(19,32)(23,33)(28,35);(2,4)(5,8,10,20)(6,12,14,25)(7,17,16,29)(9,19,21,32)(11,23,22,33)(13,24,26,34)(15,28,27,35)(18,31,30,36)

Polycyclic presentation:

2^9|d^2=i,b^a=be,c^a=cf,d^a=dg,h^a=hi,c^b=ci,f^b=fhi,e^c=eh,f^e=fi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72233533, #72234427, #72261182, …
maximal normal subgroup of	#72233533, #72234427, #72261251, …
maximal quotient of	#72232809, #72233375, #72234964, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 418974);