#263092

order	512
nilpotency class	3
isoclinism	isoclinic class of #263075
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

803 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=e,c^2=i,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,c^b=cg,g^b=gh,d^c=dh,f^c=fi,g^c=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 170288);