#247365

order	512
nilpotency class	3
isoclinism	isoclinic class of #247207
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#36493	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1361 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=hi,d^2=hi,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=dh,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, 154561);