#7196519

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996719
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	#45064	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1517 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:

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

Polycyclic presentation:

2^9|a^2=fh,c^2=fghi,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi,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, 7103715);