#242457

order	512
nilpotency class	3
isoclinism	isoclinic class of #233245
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

792 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

(5,6,9,14,11,15,10,19)(7,16,18,20,8,13,17,12)(21,22,25,29,26,30,24,28)(23,31,32,35,33,36,34,27);(1,2)(3,4)(5,7,10,17,11,8,9,18)(6,12,14,16,15,20,19,13);(2,4)(5,8)(6,13)(7,11)(9,18)(10,17)(12,19)(14,20)(15,16)(21,23)(22,27)(24,32)(25,34)(26,33)(28,31)(29,36)(30,35);(5,9,11,10)(6,14,15,19)(7,17,8,18)(12,13,20,16)(21,24,26,25)(22,28,30,29)(23,32,33,34)(27,31,35,36)

Polycyclic presentation:

2^9|a^2=ef,b^2=e,d^2=hi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,g^a=gh,c^b=cg,d^b=dh,e^c=eh,f^c=fi,g^c=gh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 149653);