#115805

order	512
nilpotency class	3
isoclinism	isoclinic class of #100051
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

812 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,e^2=hi,f^2=h,b^a=bd,c^a=ce,e^a=eh,f^a=fi,c^b=cf,e^b=ei,f^b=fh,e^c=ehi,f^c=fh,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 23001);