#511788

order	512
nilpotency class	4
isoclinism	isoclinic class of #511785
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

2440 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)(11,17)(16,29)(18,30)(19,33)(20,26)(21,27)(22,28)(23,31)(24,32)(25,34)(35,36);(5,7,10,18)(6,11,14,24)(8,19,20,16)(9,17,22,32)(12,25,26,23)(13,15,28,30)(21,34,35,31)(27,33,36,29);(8,20)(9,22)(12,26)(16,19)(17,32)(23,25)(27,36)(29,33);(2,4)(5,8,10,20)(6,12,14,26)(7,16,18,19)(9,21,22,35)(11,23,24,25)(13,27,28,36)(15,29,30,33)(17,31,32,34)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72233533, #72280213, #72312322, …
maximal normal subgroup of	#72233533, #72280213, #72312322, …
maximal quotient of	#72233418, #72234972

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 418984);