#276745

order	512
nilpotency class	3
isoclinism	isoclinic class of #268763
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

522 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:

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

Polycyclic presentation:

2^9|a^2=efh,c^2=hi,d^2=ei,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=dh,g^b=gi,f^c=fh,g^c=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 183941);