#284374

order	512
nilpotency class	3
isoclinism	isoclinic class of #264123
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

730 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)(7,11)(8,12)(13,14,17,15,18,21,19,22)(16,24,28,25,27,23,26,20);(1,2)(3,4)(5,6,9,7)(8,11,12,10)(13,15,17,21,18,22,19,14)(16,25,28,23,27,20,26,24);(2,4)(13,16,18,27)(14,20,21,25)(15,23,22,24)(17,26,19,28);(5,7,9,6)(8,10,12,11)(13,17,18,19)(14,15,21,22)(16,26,27,28)(20,23,25,24)

Polycyclic presentation:

2^9|a^2=f,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=dei,e^a=eh,c^b=cg,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, 191570);