#186662

order	512
nilpotency class	3
isoclinism	isoclinic class of #18618
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#3519	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2716 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)(5,6,10,12,9,7,8,11)(13,14)(15,23)(16,21)(17,20)(18,22)(19,25)(24,27)(26,28);(2,4)(8,10)(11,12)(13,15)(14,19)(16,26)(17,25)(18,24)(20,27)(21,23)(22,28);(5,7)(6,9)(8,12)(10,11)(14,20)(16,17)(19,27)(21,22)(23,28)(25,26);(5,8,9,10)(6,11,7,12)(15,24)(19,28)(23,27)(25,26)

Polycyclic presentation:

2^9|a^2=g,d^2=hi,e^2=i,f^2=h,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=dh,e^b=ei,f^b=fh,g^b=ghi,e^c=eh,f^c=fh

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, 93858);