#366271

order	512
nilpotency class	3
isoclinism	isoclinic class of #365053
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2363
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#13990
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

853 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)(3,13,17,24)(4,18)(5,10,6,11)(8,16,25,15)(9,20)(14,26)(19,23)(21,28)(22,27)(29,30)(31,32);(1,3,6,16,7,17,5,15)(2,8,10,24,12,25,11,13)(4,19,20,31,22,14,21,30)(9,26,18,32,28,23,27,29);(1,4,5,20,7,22,6,21)(2,9,11,18,12,28,10,27)(3,14,15,30,17,19,16,31)(8,23,13,29,25,26,24,32);(2,10,12,11)(8,24,25,13)(9,27,28,18)(23,32,26,29)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 273467);