#247423

order	512
nilpotency class	3
isoclinism	isoclinic class of #247367
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#3205	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1580 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=i,c^2=i,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,d^b=di,e^b=eh,f^c=fi,e^d=eh

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 154619);