#121888

order	512
nilpotency class	3
isoclinism	isoclinic class of #10637
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#397
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

518 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=dei,c^2=dei,d^2=hi,e^2=i,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,d^b=di,e^b=ei,f^b=fi,g^b=gi,d^c=dhi,e^c=ehi,f^c=fi,g^c=gi

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, 29084);