#512176

order	512
nilpotency class	4
isoclinism	isoclinic class of #511747
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₁₆	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

660 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=e,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,g^a=gh,h^a=hi,c^b=ci,g^d=gh,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419372);