#382093

order	512
nilpotency class	3
isoclinism	isoclinic class of #219508
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2328
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

845 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)(3,7)(4,5)(6,8)(9,10,16,20,15,21,14,19)(11,22,25,13,18,31,24,32)(12,17,28,35,29,36,33,37)(23,27,34,30,26,39,38,40);(2,5)(6,7)(9,11,15,18)(10,13,21,32)(12,26,29,23)(14,25,16,24)(17,30,36,40)(19,31,20,22)(27,37,39,35)(28,34,33,38);(1,3,4,8)(2,6,5,7)(9,12)(10,17)(11,23)(13,30)(14,33)(15,29)(16,28)(18,26)(19,37)(20,35)(21,36)(22,27)(24,38)(25,34)(31,39)(32,40);(3,8)(6,7)(9,13)(10,18)(11,21)(12,27,29,39)(14,22)(15,32)(16,31)(17,34,36,38)(19,25)(20,24)(23,35,26,37)(28,30,33,40)

Polycyclic presentation:

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

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, 289289);