#411108

order	512
nilpotency class	3
isoclinism	isoclinic class of #400057
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2631
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

758 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224750, #72224756, #72224758
maximal normal subgroup of	#72224750, #72224756, #72224758

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 318304);