#352333

order	512
nilpotency class	3
isoclinism	isoclinic class of #352320
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

615 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ei,c^2=fi,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=ch,d^b=di,g^d=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, 259529);