#394058

order	512
nilpotency class	3
isoclinism	isoclinic class of #394058
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂⁵	C₂⁴
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

12629 conjugacy classes of subgroups.

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 32

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

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 301254);