#509970

order	512
nilpotency class	4
isoclinism	isoclinic class of #509081
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₁₆	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

580 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=ei,c^2=e,d^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,f^c=fh,h^c=hi,f^d=fh,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343483, #72343833, #72343840
maximal normal subgroup of	#72343483, #72343833, #72343840

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 417166);