#215716

order	512
nilpotency class	3
isoclinism	isoclinic class of #163726
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

828 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=eh,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,d^b=di,f^c=fi,g^c=ghi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 122912);