#502091

order	512
nilpotency class	4
isoclinism	isoclinic class of #499769
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

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

759 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)(7,14)(8,10)(9,13)(11,16)(12,17)(15,22)(18,31)(19,26)(20,29)(21,32)(23,27)(24,25)(28,30)(33,34)(35,36);(1,3,4,2)(6,13)(7,18)(9,11)(10,15)(14,19)(16,17)(20,36)(21,33)(22,24)(25,27)(26,28)(29,31)(30,34)(32,35);(2,3)(5,7,12,21)(6,14,17,32)(8,19,23,35)(9,20,11,28)(10,26,27,36)(13,29,16,30)(15,18,25,33)(22,31,24,34);(5,8)(6,15)(7,19)(9,24)(10,13)(11,22)(12,23)(14,18)(16,27)(17,25)(20,34)(21,35)(26,29)(28,31)(30,36)(32,33)

Polycyclic presentation:

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

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, 409287);