#318873

order	512
nilpotency class	3
isoclinism	isoclinic class of #262883
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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

520 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)(11,17)(12,20)(14,22)(16,23)(18,24)(19,21)(25,26,28,33)(27,35,37,31)(29,32,30,34)(36,39,38,40);(1,3,5,4)(2,6,8,7)(9,10)(11,18)(12,16)(13,15)(14,19)(17,24)(20,23)(21,22)(25,27,29,36,28,37,30,38)(26,31,32,40,33,35,34,39);(9,11,13,19)(10,14,15,18)(12,21,20,17)(16,24,23,22)(26,32)(27,36)(29,30)(31,35)(33,34)(37,38);(1,4,5,3)(2,7,8,6)(10,15)(14,18)(16,23)(22,24)(27,37)(31,35)(36,38)(39,40)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 226069);