#284342

order	512
nilpotency class	3
isoclinism	isoclinic class of #264491
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

738 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=fh,c^2=i,d^2=hi,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dhi,e^b=eh,f^c=fi,g^c=gi,e^d=eh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 191538);