#508993

order	512
nilpotency class	4
isoclinism	isoclinic class of #508304
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

673 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=g,d^2=fi,g^2=hi,h^2=i,b^a=be,c^a=cf,c^b=cg,d^b=di,d^c=de,g^c=gh,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343806, #72343814, #72343818
maximal normal subgroup of	#72343806, #72343814, #72343818

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416189);