#476772

order	512
nilpotency class	3
isoclinism	isoclinic class of #476092
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

909 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

(1,2)(3,4)(5,6,8,10)(7,9,11,12)(13,14,19,23)(15,24,22,18)(16,25,30,27)(17,31,34,26)(20,35,37,29)(21,38,40,33)(28,41,43,36)(32,39,44,42);(5,7)(6,9)(8,11)(10,12)(13,15,18,14,19,22,24,23)(16,27,29,20,30,25,35,37)(17,31,33,40,34,26,38,21)(28,39,32,41,43,42,44,36);(6,10)(9,12)(13,16,19,30)(14,20,23,37)(15,25,22,27)(17,32,34,44)(18,35,24,29)(21,39,40,42)(26,41,31,36)(28,38,43,33);(2,4)(7,11)(9,12)(13,17,24,38,19,34,18,33)(14,21,15,26,23,40,22,31)(16,28,35,44,30,43,29,32)(20,36,27,42,37,41,25,39)

Polycyclic presentation:

2^9|a^2=ef,b^2=e,c^2=i,d^2=ei,e^2=i,f^2=i,h^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,d^b=def,g^b=gi,d^c=de,e^c=ei,f^c=fi,h^c=hi

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, 383968);