#6371671

order	512
nilpotency class	3
isoclinism	isoclinic class of #60470
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3165 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fhi,c^2=f,f^2=i,b^a=bf,c^a=cg,f^a=fi,g^c=gi,e^d=ei

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, 6278867);