#479767

order	512
nilpotency class	3
isoclinism	isoclinic class of #477321
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2830
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17238
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

613 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 386963);