#479851

order	512
nilpotency class	3
isoclinism	isoclinic class of #477321
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2830
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17279
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

512 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)(3,6)(4,8)(5,13,23,28)(7,11)(9,22,19,21)(10,14)(12,17)(15,29,30,27)(16,25)(18,26)(20,32,31,24);(1,3,6,11,7,10,14,2)(4,8,17,12,18,26,16,25)(5,19,22,28,23,9,21,13)(15,29,20,32,30,27,31,24);(1,4,14,16,7,18,6,17)(2,8,3,12,11,26,10,25)(5,20,21,15,23,31,22,30)(9,27,13,24,19,29,28,32);(1,5,14,21,7,23,6,22)(2,9,3,13,11,19,10,28)(4,15,16,31,18,30,17,20)(8,24,12,29,26,32,25,27)

Polycyclic presentation:

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

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, 387047);