#217806

order	512
nilpotency class	3
isoclinism	isoclinic class of #30412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×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	C₂×C₄×D₄
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

638 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)(11,17)(12,20)(14,22)(16,23)(18,24)(19,21)(25,26,31,33,29,28,30,38)(27,34,37,40,35,32,36,39);(1,3)(2,5)(4,8)(6,7)(9,10)(11,18)(12,16)(13,15)(14,19)(17,24)(20,23)(21,22)(25,27,31,37,29,35,30,36)(26,32,33,39,28,34,38,40);(3,8)(5,7)(9,11,13,19)(10,14,15,18)(12,21,20,17)(16,24,23,22)(25,28,29,26)(27,34,35,32)(30,33,31,38)(36,39,37,40);(2,6)(5,7)(10,15)(14,18)(16,23)(22,24)(26,28)(32,34)(33,38)(39,40)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,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, 125002);