#461811

order	512
nilpotency class	3
isoclinism	isoclinic class of #455826
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2631
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

783 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:

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

Polycyclic presentation:

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

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, 369007);