#217808

order	512
nilpotency class	3
isoclinism	isoclinic class of #30412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂×C₄²

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	C₂×C₄×D₄
prime radical	#23567	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

768 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)(3,7)(4,6)(5,8)(9,10,12,16)(11,18,19,14)(13,15,21,17)(20,23,24,22)(25,26,30,33,29,28,35,32)(27,34,37,39,36,31,38,40);(1,3)(2,5)(4,8)(6,7)(9,11)(10,14)(12,19)(13,20)(15,22)(16,18)(17,23)(21,24)(25,27,35,38,29,36,30,37)(26,31,32,39,28,34,33,40);(3,8)(5,7)(10,15)(13,21)(14,22)(16,17)(18,23)(20,24)(25,28)(26,29)(27,34)(30,32)(31,36)(33,35)(37,39)(38,40);(2,6)(5,7)(10,16)(11,19)(15,17)(20,24)(26,28)(31,34)(32,33)(39,40)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ceh,d^b=dhi,d^c=di,f^c=fh

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, 125004);