#504576

order	512
nilpotency class	4
isoclinism	isoclinic class of #499724
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

573 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);(2,5)(6,8)(18,22)(19,25)(20,27)(23,30)(24,31)(26,29)(28,32);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,18)(19,26)(20,29)(21,23)(22,25)(24,27)(28,31)(30,32);(2,5)(6,8)(9,11,12,15)(10,14,13,16)(18,23)(22,30)(24,29)(26,31)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343652, #72343816, #72343840
maximal normal subgroup of	#72343652, #72343816, #72343840

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 411772);