#493427

order	512
nilpotency class	4
isoclinism	isoclinic class of #480268
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	C₂×#3485	C₂
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×D₄ ◁ C₂²×H(5,𝔽₂) ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

3460 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)(9,21,22,13)(10,14)(11,17,24,31)(16,27)(18,28)(19,25)(20,33,32,26)(23,30,34,36)(29,35);(5,7,10,18)(6,11,14,24)(8,16,19,29)(9,17,22,31)(12,23,25,34)(13,15,21,28)(20,30,32,36)(26,27,33,35);(2,4)(5,8)(6,12)(7,16)(9,20)(10,19)(11,23)(13,26)(14,25)(15,27)(17,30)(18,29)(21,33)(22,32)(24,34)(28,35)(31,36);(8,19)(12,25)(16,29)(20,32)(23,34)(26,33)(27,35)(30,36)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72233533, #72233663, #72240225, …
maximal normal subgroup of	#72233533, #72233663, #72240225, …
maximal quotient of	#72233501, #72233629, #72234967, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400623);