#380288

order	512
nilpotency class	3
isoclinism	isoclinic class of #29634
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

2238 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=f,b^a=be,c^a=cf,g^a=gi,c^b=cg,f^b=fi,d^c=de,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 287484);