#372785

order	512
nilpotency class	3
isoclinism	isoclinic class of #25140
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1444 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

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, 279981);