#6442588

order	512
nilpotency class	3
isoclinism	isoclinic class of #6442587
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4572 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6349784);