#283147

order	512
nilpotency class	3
isoclinism	isoclinic class of #249463
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

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

1002 conjugacy classes of subgroups.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 20 points:

(1,2,4,3)(5,6)(7,12)(8,10)(9,11);(1,3,4,2)(6,10)(7,8)(11,12)(13,14)(15,17)(16,19)(18,20);(2,3)(6,11)(10,12)(14,17)(15,20)(18,19);(5,7)(8,9)(10,12)

Polycyclic presentation:

2^9|a^2=fh,b^2=fh,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,e^b=eh,g^b=gi,g^c=gi,e^d=eh

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 190343);