#448227

order	512
nilpotency class	3
isoclinism	isoclinic class of #419458
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2709
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15907
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

695 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 355423);