#414747

order	512
nilpotency class	3
isoclinism	isoclinic class of #394267
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

2568 conjugacy classes of subgroups.

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

A minimal faithful permutation representation — 4 generators on 20 points:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 321943);