#272221

order	512
nilpotency class	3
isoclinism	isoclinic class of #272155
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#61067	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1085 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)(3,5)(4,7)(6,8)(9,10,13,18,15,20,14,19)(11,16,22,31,24,33,23,32)(12,25,29,17,28,36,27,35)(21,37,26,30,39,34,38,40);(2,5)(3,8)(6,7)(9,11)(10,16)(12,26)(13,22)(14,23)(15,24)(17,34)(18,31)(19,32)(20,33)(21,27)(25,30)(28,38)(29,39)(35,37)(36,40);(9,12)(10,17)(11,21)(13,27)(14,29)(15,28)(16,30)(18,25)(19,36)(20,35)(22,38)(23,26)(24,39)(31,37)(32,34)(33,40);(2,5)(3,8)(6,7)(9,13,15,14)(10,18,20,19)(11,22,24,23)(12,27,28,29)(16,31,33,32)(17,25,35,36)(21,38,39,26)(30,37,40,34)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 179417);