#194515

order	512
nilpotency class	3
isoclinism	isoclinic class of #19014
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1500 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,e^2=h,f^2=hi,g^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fh,d^b=dh,e^b=eh,f^b=fi,g^b=gh,e^c=ei,f^c=fhi,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, 101711);