#468932

order	512
nilpotency class	3
isoclinism	isoclinic class of #455561
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2622
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

750 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

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, 376128);