#381123

order	512
nilpotency class	3
isoclinism	isoclinic class of #214664
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1256 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:

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

Polycyclic presentation:

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

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, 288319);