#111869

order	512
nilpotency class	3
isoclinism	isoclinic class of #10440
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#394
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂²×#459	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1330 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=i,e^2=h,f^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,d^b=di,e^b=eh,f^b=fh,g^b=ghi,e^c=eh,f^c=fh,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, 19065);