#104822

order	512
nilpotency class	3
isoclinism	isoclinic class of #102905
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	#120
commutator, Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#17068	C₂
LCS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

969 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 28 points:

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

Polycyclic presentation:

2^9|c^2=dg,e^2=g,f^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fh,c^b=cf,e^b=eh,f^b=fi,e^c=eg,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72217734
maximal normal subgroup of	#72217734
maximal quotient of	#72216328, #72216354

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 12018);