#295272

order	512
nilpotency class	3
isoclinism	isoclinic class of #268211
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2819	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

789 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)(3,7)(4,5)(6,8)(9,10)(11,15)(12,16)(13,14)(17,18,21,26)(19,28,30,24)(20,32,34,44)(22,36,23,27)(25,38,43,35)(29,46,47,42)(31,40,37,41)(33,39,48,45);(2,5)(6,7)(11,16)(12,15)(17,19)(18,24)(20,33)(21,30)(22,31)(23,37)(25,42)(26,28)(27,40)(29,35)(32,45)(34,48)(36,41)(38,47)(39,44)(43,46);(9,11,15,10,13,16,12,14)(17,20,23,38,21,34,22,35)(18,25,36,44,26,43,27,32)(19,29,37,33,30,47,31,48)(24,39,41,46,28,45,40,42);(1,3,4,8)(2,6,5,7)(18,26)(20,34)(24,28)(27,36)(29,47)(33,48)(35,38)(40,41)

Polycyclic presentation:

2^9|a^2=hi,c^2=fh,d^2=ehi,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=ghi,d^c=dhi,g^c=gh

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 202468);