#219056

order	512
nilpotency class	3
isoclinism	isoclinic class of #30396
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄

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	#1999	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

685 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)(12,19)(14,20)(16,22)(18,23)(21,24)(25,26,30,34,29,32,36,33)(27,35,28,38,37,31,39,40);(1,3)(2,5)(4,8)(6,7)(9,10,13,15)(11,14,18,21)(12,16,19,22)(17,20,23,24)(25,27,36,39,29,37,30,28)(26,31,33,38,32,35,34,40);(3,8)(5,7)(9,11,12,17,13,18,19,23)(10,14,16,20,15,21,22,24)(25,28,30,37,29,39,36,27)(26,31,33,38,32,35,34,40);(2,6)(5,7)(10,15)(11,18)(16,22)(17,23)(26,32)(31,35)(33,34)(38,40)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=f,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	#72343768, #72343881, #72343892
maximal normal subgroup of	#72343768, #72343881, #72343892

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 126252);