#397885

order	512
nilpotency class	3
isoclinism	isoclinic class of #394177
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2410 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=f,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 305081);