#7576530

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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	#23850	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1693 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7483726);