#502093

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	#25805	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

783 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,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, 409289);