#371418

order	512
nilpotency class	3
isoclinism	isoclinic class of #24468
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#120
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1057 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

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, 278614);