#279609

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, prime radical	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

519 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=efh,b^2=e,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, 186805);