#109697

order	512
nilpotency class	3
isoclinism	isoclinic class of #9183
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

672 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,d^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,g^a=ghi,d^b=di,e^b=ehi,d^c=dhi

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, 16893);