#374670

order	512
nilpotency class	3
isoclinism	isoclinic class of #25140
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×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	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1274 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)(3,4)(5,6,8,7)(9,10,14,19,15,20,13,18)(11,16,24,33,23,32,22,31)(12,25,27,17,29,35,28,36)(21,37,38,30,26,40,39,34);(2,4)(9,11,13,22,15,23,14,24)(10,16,18,31,20,32,19,33)(12,26,28,38,29,21,27,39)(17,34,25,40,36,30,35,37);(5,7,8,6)(9,12)(10,17)(11,21)(13,27)(14,28)(15,29)(16,30)(18,35)(19,25)(20,36)(22,38)(23,26)(24,39)(31,40)(32,34)(33,37);(2,4)(9,13,15,14)(10,18,20,19)(11,22,23,24)(12,27,29,28)(16,31,32,33)(17,35,36,25)(21,38,26,39)(30,40,34,37)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=fh,d^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gi,f^c=fi,h^c=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, 281866);