#7555185

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35689	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1655 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,d^2=fg,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=dh,h^b=hi,e^c=ei,h^d=hi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313550, #72313586, #72313939
maximal normal subgroup of	#72313550, #72313586, #72313939

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7462381);