#153971

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄³

Aliases

No aliases for this group.

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁵	C₂⁴
commutator	C₂³	C₂²×C₄²
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

3746 conjugacy classes of subgroups.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,d^2=e,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 61167);