#500591

order	512
nilpotency class	4
isoclinism	isoclinic class of #500591
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3253
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#24152
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

832 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)(3,12)(4,17)(5,22)(6,23)(7,11)(8,13)(9,15)(10,14)(16,31)(18,28)(19,30)(20,26)(21,25)(24,27)(29,32);(1,3,5,15,8,16,7,6)(2,9,11,12,13,23,22,31)(4,18,20,14,21,24,19,32)(10,26,28,17,29,30,27,25);(1,4)(2,10)(3,14)(5,19)(6,24)(7,20)(8,21)(9,25)(11,27)(12,30)(13,29)(15,18)(16,32)(17,23)(22,28)(26,31);(1,3,5,15,8,16,7,6)(2,9,11,12,13,23,22,31)(4,14,19,18,21,32,20,24)(10,25,27,30,29,17,28,26)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72219593, #72219597, #72219599, …
maximal normal subgroup of	#72219593, #72219597, #72219599, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407787);