#7329149

order	512
nilpotency class	3
isoclinism	isoclinic class of #7324500
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	#1999	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1439 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7236345);