#355487

order	512
nilpotency class	3
isoclinism	isoclinic class of #21782
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1597 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=fgi,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=ce,g^b=gi,e^c=ei,f^d=fi,g^d=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 262683);