#376566

order	512
nilpotency class	3
isoclinism	isoclinic class of #25140
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#121
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1174 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(5,6,9,15)(7,16)(8,21)(10,14,17,31)(11,20)(12,25)(13,23)(18,33,34,27)(19,30)(22,29)(24,28)(26,36,32,35);(1,2)(3,4)(5,7,17,19,9,20,10,24)(6,11,14,30,15,16,31,28)(8,18,22,36,23,34,25,35)(12,26,13,27,29,32,21,33);(2,4)(5,8)(6,12)(7,18)(9,23)(10,25)(11,26)(13,14)(15,29)(16,32)(17,22)(19,36)(20,34)(21,31)(24,35)(27,30)(28,33);(5,8)(6,13,15,21)(7,18)(9,23)(10,25)(11,27,16,33)(12,14,29,31)(17,22)(19,36)(20,34)(24,35)(26,30,32,28)

Polycyclic presentation:

2^9|a^2=h,b^2=ef,d^2=eh,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,e^b=ei,f^b=fi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 283762);