#119225

order	512
nilpotency class	3
isoclinism	isoclinic class of #99947
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

548 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 26421);