#121742

order	512
nilpotency class	3
isoclinism	isoclinic class of #10304
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	#127
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

746 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=dehi,c^2=de,d^2=h,e^2=h,f^2=i,g^2=h,b^a=bd,c^a=ce,c^b=cf,d^b=dh,e^b=eh,f^b=fi,g^b=gh,d^c=dh,e^c=eh,f^c=fi,g^c=gh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28938);