#456879

order	512
nilpotency class	3
isoclinism	isoclinic class of #454109
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2670
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16287
prime radical	#23412	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1136 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)(3,14)(4,18,21,9)(5,22)(6,11,15,13)(8,24)(10,23)(16,29)(17,28)(19,31)(20,27,32,30)(25,26);(1,3,15,16,7,17,6,24)(2,8,13,14,12,29,11,28)(4,19,32,23,21,25,20,5)(9,10,27,26,18,22,30,31);(1,4,6,20,7,21,15,32)(2,9,11,30,12,18,13,27)(3,5,16,19,17,23,24,25)(8,26,14,22,29,31,28,10);(1,5)(2,10)(3,4)(6,25)(7,23)(8,27)(9,28)(11,31)(12,22)(13,26)(14,18)(15,19)(16,32)(17,21)(20,24)(29,30)

Polycyclic presentation:

2^9|a^2=f,b^2=ef,c^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,h^c=hi,e^d=ei,f^d=fi,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, 364075);