#6452025

order	512
nilpotency class	3
isoclinism	isoclinic class of #24398
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×#121
prime radical	C₂×#3002	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2673 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2)(3,8)(4,6)(5,7)(9,10,13,18,15,20,14,19)(11,21,24,32,25,31,23,16)(12,17,27,34,28,35,29,36)(22,33,26,39,38,40,37,30);(9,11)(10,16)(12,26)(13,23)(14,24)(15,25)(17,33)(18,31)(19,21)(20,32)(22,27)(28,37)(29,38)(30,34)(35,40)(36,39);(1,3,5,4)(2,6,7,8)(9,12)(10,17)(11,22)(13,27)(14,29)(15,28)(16,30)(18,34)(19,36)(20,35)(21,33)(23,37)(24,26)(25,38)(31,40)(32,39);(1,4,5,3)(2,6,7,8)(9,12)(10,17)(11,22)(13,27)(14,29)(15,28)(16,30)(18,34)(19,36)(20,35)(21,33)(23,37)(24,26)(25,38)(31,40)(32,39);(9,13,15,14)(10,18,20,19)(11,23,25,24)(12,27,28,29)(16,31,32,21)(17,34,35,36)(22,37,38,26)(30,40,39,33)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6359221);