#515341

order	512
nilpotency class	2
isoclinism	isoclinic class of #35346
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

4118 conjugacy classes of subgroups.

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

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=g,d^2=f,e^2=f,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=df

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 422537);