#508860

order	512
nilpotency class	4
isoclinism	isoclinic class of #508292
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 ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

609 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)(11,13)(12,14)(15,16)(17,18,23,27,21,20,25,31,22,26,30,32,24,19,28,29);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(17,19,23,29,21,18,25,27,22,20,30,31,24,26,28,32);(1,3)(2,6)(4,7)(5,8)(10,14)(13,16)(17,20)(18,21)(19,22)(23,27)(24,26)(25,29)(28,31)(30,32);(3,7)(6,8)(17,21,22,24)(18,20,26,19)(23,25,30,28)(27,31,32,29)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343806, #72343824, #72343833
maximal normal subgroup of	#72343806, #72343824, #72343833

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416056);