#121971

order	512
nilpotency class	3
isoclinism	isoclinic class of #10012
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#127
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#399
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

528 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=deh,c^2=de,d^2=h,e^2=i,f^2=h,g^2=h,b^a=bd,c^a=ce,f^a=fhi,c^b=cf,d^b=dh,e^b=eh,f^b=fh,g^b=gh,d^c=di,e^c=ei,f^c=fh,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29167);