#234042

order	512
nilpotency class	3
isoclinism	isoclinic class of #234028
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

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

1438 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|d^2=hi,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=cg,d^b=di,e^b=eh,d^c=dh,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 141238);