#235743

order	512
nilpotency class	3
isoclinism	isoclinic class of #233984
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
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1544 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 142939);