#263658

order	512
nilpotency class	3
isoclinism	isoclinic class of #263635
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
prime radical	C₂×#41	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

804 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 170854);