#248445

order	512
nilpotency class	3
isoclinism	isoclinic class of #248359
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#61713	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1100 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)(3,10)(4,7)(5,9)(6,11)(8,14)(12,16)(13,15)(18,24)(19,26)(20,28)(21,22)(29,30)(31,32);(1,3,5,12)(2,6,9,15)(4,13,14,11)(7,10,8,16)(17,18)(19,23)(20,25)(21,26)(22,24)(27,29)(28,31)(30,32);(6,15)(10,16)(17,19,23,18)(20,25,29,27)(21,26,22,24)(28,32,30,31);(2,7)(4,14)(6,10)(8,9)(11,13)(15,16)(17,20,23,29)(18,25,19,27)(21,30,22,28)(24,31,26,32)

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 155641);