#497395

order	512
nilpotency class	4
isoclinism	isoclinic class of #481415
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#3280	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1153 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:

(2,4)(5,6,9,14,10,16,24,15)(7,11,18,27,19,29,35,28)(8,20,22,13,23,12,21,30)(17,31,33,26,34,25,32,36);(1,2)(3,4)(5,7,10,19)(6,11,16,29)(8,17,23,34)(9,18,24,35)(12,25,20,31)(13,26,30,36)(14,27,15,28)(21,32,22,33);(6,12)(8,21)(9,24)(11,25)(13,14)(15,30)(16,20)(17,32)(18,35)(22,23)(26,27)(28,36)(29,31)(33,34);(2,4)

Polycyclic presentation:

2^9|a^2=g,b^2=i,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^b=de,f^c=fhi,g^c=gi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250863, #72250867, #72250908, …
maximal normal subgroup of	#72250863, #72250867, #72250908, …
maximal quotient of	#72287660, #72309423, #72309688, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404591);