#400698

order	512
nilpotency class	3
isoclinism	isoclinic class of #393934
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂×#2021	C₂
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

2452 conjugacy classes of subgroups.

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 16 points:

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

Polycyclic presentation:

2^9|b^2=ef,d^2=ef,b^a=be,c^a=cf,d^a=dh,c^b=cg,h^b=hi,h^c=hi,e^d=ei,f^d=fi

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 307894);