#366718

order	512
nilpotency class	3
isoclinism	isoclinic class of #24289
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2233 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 273914);