#497837

order	512
nilpotency class	4
isoclinism	isoclinic class of #483463
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#106
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₄	SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

626 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286580, #72286595, D₄×#2155, …
maximal normal subgroup of	#72286580, #72286595, D₄×#2155, …
maximal quotient of	#72287034, #72287405, #72309475, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405033);