#216001

order	512
nilpotency class	3
isoclinism	isoclinic class of #30396
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

793 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=e,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,d^b=dhi,f^c=fh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 123197);