#239028

order	512
nilpotency class	3
isoclinism	isoclinic class of #239025
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	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

814 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,10)(4,14)(5,8)(6,9)(7,12)(11,16)(13,15)(17,18,23,28)(19,29,33,40)(20,34,37,25)(21,27,22,26)(24,31,41,32)(30,45,47,42)(35,48,46,39)(36,43,38,44);(1,2)(3,10)(4,8)(5,14)(6,9)(7,13)(11,16)(12,15)(17,19)(18,24)(20,35)(21,31)(22,32)(23,33)(25,42)(26,40)(27,29)(28,41)(30,38)(34,45)(36,47)(37,46)(39,44)(43,48);(1,3)(2,7)(4,13)(5,12)(6,11)(8,16)(9,15)(10,14)(17,20,21,36,23,37,22,38)(18,25,26,43,28,34,27,44)(19,30,32,46,33,47,31,35)(24,39,29,45,41,48,40,42);(3,11)(7,15)(10,16)(12,13)(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,36,37,38)(24,40,41,29)(25,43,34,44)(30,35,47,46)(39,42,48,45)

Polycyclic presentation:

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

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, 146224);