#265224

order	512
nilpotency class	3
isoclinism	isoclinic class of #265163
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	C₂²×D₄	#41
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

847 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)(3,11)(4,7)(5,10)(6,9)(8,14)(12,15)(13,16)(17,18,23,28)(19,29,33,40)(20,34,37,44)(21,27,22,26)(24,32,41,31)(25,38,43,36)(30,42,35,39)(45,47,48,46);(2,7)(3,12)(5,6)(8,11)(13,14)(15,16)(17,19,22,32,23,33,21,31)(18,24,27,40,28,41,26,29)(20,30,38,47,37,35,36,46)(25,39,44,48,43,42,34,45);(1,3,4,13)(2,8,7,15)(5,12,6,14)(9,16,10,11)(17,20,23,37)(18,25,28,43)(19,30,33,35)(21,36,22,38)(24,39,41,42)(26,34,27,44)(29,45,40,48)(31,46,32,47);(2,7)(3,13)(9,10)(12,14)(17,19,21,31,23,33,22,32)(18,24,26,29,28,41,27,40)(20,35,36,47,37,30,38,46)(25,42,34,48,43,39,44,45)

Polycyclic presentation:

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

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, 172420);