#376005

order	512
nilpotency class	3
isoclinism	isoclinic class of #23876
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#120
prime radical	#3449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 283201);