#492960

order	512
nilpotency class	3
isoclinism	isoclinic class of #20671
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#432
prime radical	C₂⁵	M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1234 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2)(3,4)(5,6)(7,17,20,18)(8,14)(9,15)(10,16)(11,13)(12,19,27,22)(21,31,34,32)(23,29)(24,26)(25,30)(28,33,36,35);(2,4)(5,7,9,21,10,19,25,33,11,20,26,34,8,22,23,35)(6,12,15,28,16,17,30,31,13,27,24,36,14,18,29,32);(5,6)(7,18,20,17)(8,14)(9,24)(10,16)(11,13)(12,22,27,19)(15,26)(21,31,34,32)(23,30)(25,29)(28,33,36,35);(6,13)(12,27)(14,16)(15,24)(17,18)(28,36)(29,30)(31,32)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,e^2=i,g^2=h,h^2=i,b^a=be,d^a=di,c^b=ch,e^b=ei,f^b=fi,d^c=di,g^c=gi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400156);