#457602

order	512
nilpotency class	3
isoclinism	isoclinic class of #453948
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1531 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 364798);