#476265

order	512
nilpotency class	3
isoclinism	isoclinic class of #476092
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1307 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)(7,10)(8,11)(9,12)(13,14,17,21,19,24,18,23)(15,25,27,36,29,34,28,20)(16,30,31,38,33,37,32,22)(26,35,40,43,42,44,41,39);(5,7)(6,9)(8,12)(10,11)(13,15,19,29)(14,20,24,36)(16,26,33,42)(17,28,18,27)(21,34,23,25)(22,35,38,44)(30,39,37,43)(31,41,32,40);(5,7)(6,10)(8,12)(9,11)(14,21)(16,31)(17,18)(20,34)(23,24)(25,36)(26,40)(27,28)(30,37)(32,33)(35,44)(41,42);(2,4)(7,12)(9,10)(13,16)(14,22)(15,26)(17,32)(18,31)(19,33)(20,35)(21,37)(23,30)(24,38)(25,39)(27,41)(28,40)(29,42)(34,43)(36,44)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=def,e^b=ei,f^b=fi,h^b=hi,d^c=de,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=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, 383461);