#285626

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
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

664 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=fh,c^2=hi,d^2=hi,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, 192822);