#7182406

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993875
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#60924	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1620 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fh,c^2=i,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,f^d=fi

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, 7089602);