#376062

order	512
nilpotency class	3
isoclinism	isoclinic class of #23876
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

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	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

962 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)(3,8)(4,6)(5,7)(9,10,14,20,15,21,33,19)(11,16,23,27,25,36,24,35)(12,26,29,38,30,37,28,17)(13,18,31,39,32,40,34,22);(1,3,5,4)(2,6,7,8)(9,11)(10,16)(12,18)(13,17)(14,23)(15,25)(19,35)(20,27)(21,36)(22,28)(24,33)(26,31)(29,39)(30,40)(32,38)(34,37);(1,4,5,3)(2,6,7,8)(9,12)(10,17)(11,22)(13,27)(14,29)(15,30)(16,34)(18,23)(19,37)(20,26)(21,38)(24,40)(25,39)(28,33)(31,36)(32,35);(2,7)(6,8)(9,13,14,31,15,32,33,34)(10,18,20,39,21,40,19,22)(11,17,23,26,25,38,24,37)(12,27,29,36,30,35,28,16)

Polycyclic presentation:

2^9|a^2=h,b^2=e,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, 283258);