#400045

order	512
nilpotency class	3
isoclinism	isoclinic class of #393925
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

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

1387 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)(8,9)(11,12)(13,14)(15,26)(16,32)(17,35)(18,24)(19,23)(20,25)(21,29)(22,36)(27,43)(28,39)(30,41)(31,40)(33,42)(34,37)(38,44);(2,4)(6,9)(10,12)(13,15,20,31)(14,21,25,41)(16,33,22,39)(17,28,37,42)(18,30,19,29)(23,40,24,26)(27,34,44,35)(32,43,36,38);(2,4)(13,16,20,22)(14,17,25,37)(15,27,31,44)(18,35,19,34)(21,38,41,43)(23,32,24,36)(26,42,40,28)(29,33,30,39);(5,7,8,11)(6,10,9,12)(13,17)(14,22)(15,28)(16,25)(18,32)(19,36)(20,37)(21,39)(23,34)(24,35)(26,27)(29,38)(30,43)(31,42)(33,41)(40,44)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=ef,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,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, 307241);