#285568

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	#18566	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

700 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=f,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,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, 192764);