#285514

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

636 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)(17,18,20,25,22,26,21,24)(19,27,30,28,31,32,29,23);(1,3,6,9)(2,4,11,14)(5,13,7,8)(10,16,15,12)(17,19,20,30,22,31,21,29)(18,23,25,27,26,28,24,32);(1,4,6,14)(2,8,11,13)(3,10,9,15)(5,12,7,16)(17,18)(19,27)(20,24)(21,25)(22,26)(23,30)(28,29)(31,32);(1,4,6,14)(2,9,11,3)(5,16,7,12)(8,15,13,10)(17,18)(19,28)(20,25)(21,24)(22,26)(23,31)(27,29)(30,32)

Polycyclic presentation:

2^9|a^2=f,b^2=f,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, 192710);