#285424

order	512
nilpotency class	3
isoclinism	isoclinic class of #285371
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	#18620	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

708 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)(3,6)(4,11)(5,7)(8,12)(9,15)(10,14)(13,16)(17,18,20,24,21,25,28,23)(19,22,27,29,26,30,32,31);(2,7)(3,9)(5,15)(8,11)(10,16)(12,14)(17,19,20,27,21,26,28,32)(18,22,24,29,25,30,23,31);(1,3,6,2)(4,10,13,8)(5,7,15,9)(11,12,16,14)(18,23)(19,26)(20,28)(22,29)(24,25)(30,31);(1,4,6,13)(2,8,3,10)(5,14,15,12)(7,11,9,16)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 192620);