#471161

order	512
nilpotency class	3
isoclinism	isoclinic class of #459145
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2751
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16767
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

530 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 378357);