#240816

order	512
nilpotency class	3
isoclinism	isoclinic class of #240803
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

850 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)(3,7)(4,9)(5,13)(6,10)(8,11)(12,16)(14,15)(17,18,23,26)(19,27,28,31)(20,21,25,22)(24,30,32,29);(1,3)(2,7)(4,11)(5,14)(6,12)(8,9)(10,16)(13,15)(17,19,23,28)(18,24,26,32)(20,31,25,27)(21,30,22,29);(1,3)(2,8)(4,11)(5,14)(6,12)(7,13)(9,16)(10,15)(17,20,21,26,23,25,22,18)(19,24,29,27,28,32,30,31);(1,4,6,5)(2,9,10,13)(3,11,12,14)(7,8,16,15)(19,28)(24,32)(27,31)(29,30)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=e,d^2=hi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=gi,c^b=cg,d^b=dh,e^b=eh,f^b=fi,g^b=gh,f^c=fhi

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, 148012);