#501036

order	512
nilpotency class	4
isoclinism	isoclinic class of #499769
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#3415	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

994 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)(3,4)(5,6,9,15,12,16,10,11)(7,17,19,13,21,26,20,28)(8,22,23,30,25,29,24,14)(18,32,33,36,35,27,34,31);(2,4)(6,11)(7,13)(9,10)(14,22)(15,16)(17,19)(18,27)(20,26)(21,28)(23,24)(29,30)(31,34)(32,35)(33,36);(5,7)(6,13)(8,18)(9,20)(10,19)(11,26)(12,21)(14,27)(15,17)(16,28)(22,36)(23,34)(24,33)(25,35)(29,31)(30,32);(2,4)(5,8)(6,14)(7,18)(9,24)(10,23)(11,22)(12,25)(13,27)(15,29)(16,30)(17,31)(19,34)(20,33)(21,35)(26,36)(28,32)

Polycyclic presentation:

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

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, 408232);