#283613

order	512
nilpotency class	3
isoclinism	isoclinic class of #283579
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	#18888	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

669 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:

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

Polycyclic presentation:

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

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, 190809);