#246739

order	512
nilpotency class	3
isoclinism	isoclinic class of #246591
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	#24210	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1248 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)(15,22)(16,27)(17,23)(18,21)(19,24)(20,26)(25,28);(1,2)(3,4)(5,6)(7,10)(8,12)(9,11)(13,15,19,25)(14,20,24,27)(16,18,26,17)(21,28,23,22);(2,4)(5,7,9,8)(6,10,11,12)(14,21)(15,25)(17,18)(20,22)(23,24)(27,28);(5,6)(7,10)(8,12)(9,11)(13,16,19,26)(14,22,24,28)(15,17,25,18)(20,23,27,21)

Polycyclic presentation:

2^9|b^2=i,c^2=hi,d^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^b=fi,e^c=ei,f^c=fi,e^d=eh,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 153935);