#271889

order	512
nilpotency class	3
isoclinism	isoclinic class of #271771
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#38922	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1000 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)(13,14,19,21,18,23,17,22)(15,20,24,28,25,27,16,26);(1,2)(3,4)(5,6,9,11)(7,10,8,12)(13,15)(14,20)(16,17)(18,25)(19,24)(21,28)(22,26)(23,27);(2,4)(5,7,9,8)(6,10,11,12)(13,16)(14,20)(15,17)(18,24)(19,25)(21,26)(22,28)(23,27);(5,6)(7,10)(8,12)(9,11)(13,17,18,19)(14,21,23,22)(15,24,25,16)(20,26,27,28)

Polycyclic presentation:

2^9|a^2=fhi,b^2=h,c^2=h,d^2=i,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,d^b=dhi,e^b=eh,f^c=fi,e^d=eh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 179085);