#417711

order	512
nilpotency class	3
isoclinism	isoclinic class of #417603
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3013	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2351 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,d^2=i,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,h^d=hi

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, 324907);