#396932

order	512
nilpotency class	3
isoclinism	isoclinic class of #395240
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	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1072 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)(7,9)(8,10)(11,12)(13,14)(15,25)(16,29)(17,33)(18,24)(19,23)(20,27)(21,28)(22,30)(26,40)(31,43)(32,38)(34,36)(35,42)(37,44)(39,41);(5,7,8,11)(6,9,10,12)(13,15,19,34,18,28,17,27)(14,20,33,21,24,36,23,25)(16,26,32,44,31,42,30,41)(22,35,43,37,38,40,29,39);(6,10)(9,12)(13,15)(14,21)(16,26)(17,27)(18,28)(19,34)(20,23)(22,37)(24,25)(29,35)(30,41)(31,42)(32,44)(33,36)(38,39)(40,43);(2,4)(13,16,17,30,18,31,19,32)(14,22,23,29,24,38,33,43)(15,26,27,41,28,42,34,44)(20,35,25,39,36,40,21,37)

Polycyclic presentation:

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

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, 304128);