#397006

order	512
nilpotency class	3
isoclinism	isoclinic class of #393925
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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1046 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:

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

Polycyclic presentation:

2^9|b^2=f,c^2=ei,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,g^b=gi,g^c=gi,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, 304202);