#120002

order	512
nilpotency class	3
isoclinism	isoclinic class of #9091
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

610 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 28 points:

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

Polycyclic presentation:

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

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, 27198);