#263461

order	512
nilpotency class	3
isoclinism	isoclinic class of #263459
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×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)(3,11)(4,10)(5,9)(6,8)(7,13)(12,16)(14,15)(18,24)(19,26)(20,28)(21,22)(25,27)(29,30);(1,3,5,13)(2,7,9,11)(4,12,6,14)(8,16,10,15)(17,18,21,24,23,19,22,26)(20,25,28,31,30,27,29,32);(2,8)(4,6)(7,15)(9,10)(11,16)(12,14)(17,19)(18,23)(20,25)(21,26)(22,24)(27,30)(28,31)(29,32);(1,4,5,6)(2,8,9,10)(3,12,13,14)(7,16,11,15)(17,20,22,29,23,30,21,28)(18,25,26,32,19,27,24,31)

Polycyclic presentation:

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

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, 170657);