#318420

order	512
nilpotency class	3
isoclinism	isoclinic class of #262883
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

616 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

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

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, 225616);