#110052

order	512
nilpotency class	3
isoclinism	isoclinic class of #9103
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

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 ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

816 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,e^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=ci,e^b=eh,f^b=fh,d^c=dh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17248);