#285525

order	512
nilpotency class	3
isoclinism	isoclinic class of #285499
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#18568	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

628 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fh,c^2=i,d^2=i,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,e^b=eh,g^b=ghi,f^c=fi,g^c=gh,e^d=eh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 192721);