#493392

order	512
nilpotency class	4
isoclinism	isoclinic class of #493391
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#1781
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	#62852	C₂
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×H(5,𝔽₂) ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

3429 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2)(3,11)(4,8)(5,17,18,9)(6,10)(7,13,20,27)(12,24)(14,23)(15,21)(16,29,28,22)(19,26,30,32)(25,31);(1,3)(2,7)(4,12)(5,13)(6,14)(8,19)(9,11)(10,20)(15,25)(16,26)(17,23)(18,27)(21,30)(22,24)(28,32)(29,31);(4,15)(5,18)(8,21)(9,17)(11,23)(12,25)(13,27)(19,30);(1,4,6,15)(2,8,10,21)(3,12,14,25)(5,16,18,28)(7,19,20,30)(9,22,17,29)(11,24,23,31)(13,26,27,32)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72213476, #72213538, #72213574, …
maximal normal subgroup of	#72213476, #72213538, #72213574, …
maximal quotient of	#72236640, #72236806, #72256665

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400588);