#110051

order	512
nilpotency class	3
isoclinism	isoclinic class of #1909
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

996 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17247);