#511780

order	512
nilpotency class	4
isoclinism	isoclinic class of #511779
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 ◁ C₂⁴ ◁ C₂³×D₄ ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

2472 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)(16,28)(17,29)(19,25)(20,26)(21,34)(22,27)(23,31)(24,32)(30,36)(33,35);(5,7)(6,11)(8,16)(9,18)(10,17)(12,23)(13,15)(14,24)(19,31)(20,30)(21,33)(22,32)(25,28)(26,35)(27,29)(34,36);(5,8,10,20)(6,12,14,26)(7,16,17,30)(9,21,22,19)(11,23,24,35)(13,25,27,34)(15,28,29,36)(18,33,32,31);(2,4)(7,17)(11,24)(15,29)(16,30)(18,32)(23,35)(28,36)(31,33)

Polycyclic presentation:

2^9|c^2=i,b^a=be,c^a=cf,d^a=dg,h^a=hi,d^b=di,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, #72261251, #72261638, …
maximal normal subgroup of	#72233533, #72261251, #72261638, …
maximal quotient of	#72233408, #72236937, #72281714, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 418976);