#6500031

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500028
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35000	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3564 conjugacy classes of subgroups.

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

A minimal faithful permutation representation — 5 generators on 20 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,f^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,f^b=fi,d^c=dh

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6407227);