#352398

order	512
nilpotency class	3
isoclinism	isoclinic class of #352384
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2856
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

603 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=eh,c^2=fh,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,g^a=gh,c^b=ci,d^b=di,d^c=di,g^d=gh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 259594);