#7614734

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518944
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	#28349	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1560 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)(3,6)(4,5)(7,8)(9,10,12,14)(11,15,16,13)(17,18,22,20)(19,27,28,24)(21,29,32,25)(23,26,31,30);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,28)(18,24,20,27)(21,30,32,26)(23,29,31,25);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(17,20,22,18)(19,27,28,24)(21,29,32,25)(23,30,31,26);(10,14)(13,15)(17,21,31,27,22,32,23,24)(18,25,30,19,20,29,26,28);(2,5)(6,8)(10,14)(13,15)(18,20)(21,32)(24,27)(26,30)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7521930);