#478306

order	512
nilpotency class	3
isoclinism	isoclinic class of #477321
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2830
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17139
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

788 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=def,f^b=fi,d^c=de,e^c=ei,f^c=fi,g^c=gi,e^d=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, 385502);