#498168

order	512
nilpotency class	4
isoclinism	isoclinic class of #33854
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#2193	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

740 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,e^2=hi,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,h^a=hi,d^b=deh,e^b=eh,h^b=hi,d^c=di,f^c=fi,e^d=eh,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405364);