#219058

order	512
nilpotency class	3
isoclinism	isoclinic class of #30396
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

648 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)(3,7)(4,6)(5,8)(11,17)(12,18)(14,20)(16,21)(19,23)(22,24)(25,26,31,35,29,34,30,39)(27,33,28,38,36,32,37,40);(1,3)(2,5)(4,8)(6,7)(9,10,13,15)(11,14,19,22)(12,16,18,21)(17,20,23,24)(25,27,30,37,29,36,31,28)(26,32,39,38,34,33,35,40);(3,8)(5,7)(9,11,18,23,13,19,12,17)(10,14,21,24,15,22,16,20)(25,28,31,36,29,37,30,27)(26,33,39,40,34,32,35,38);(2,6)(5,7)(10,15)(11,19)(16,21)(17,23)(26,34)(32,33)(35,39)(38,40)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=fh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cei,d^b=dhi,d^c=dhi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343703, #72343881, #72343892
maximal normal subgroup of	#72343703, #72343881, #72343892

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 126254);