#189141

order	512
nilpotency class	3
isoclinism	isoclinic class of #19420
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
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

1134 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=ch,d^b=di,f^c=fi

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, 96337);