#94741

order	512
nilpotency class	6
isoclinism	isoclinic class of #94690
automorphism	?
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²	#1415
commutator	C₂×C₃₂	C₂×C₄
Frattini, cosocle	C₄×C₃₂	C₂²
LCS	G ▷ C₂×C₃₂ ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×C₃₂ ◁ G
LECS	G ▷ C₄×C₃₂ ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

65 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 2 generators on 72 points:

(2,5)(3,7)(6,8)(9,10,12,19,32,49,67,71,15,22,34,41,39,55,25,40,37,53,28,18,44,46,62,70,68,72,66,47,14,21,33,43,13,20,30,50,26,45,63,56,36,52,61,69,64,48,60,17,16,23,11,24,27,58,65,42,38,54,29,59,35,51,31,57);(1,2,4,6)(3,8,7,5)(10,17)(11,25)(12,31)(14,26)(15,38)(16,37)(18,21)(19,48)(20,40)(22,56)(23,57)(24,51)(27,39)(28,60)(29,67)(30,33)(32,35)(34,65)(36,68)(41,45)(42,54)(43,53)(44,64)(46,47)(49,69)(50,55)(52,71)(58,59)(61,62)(63,66)(70,72)

Polycyclic presentation:

2^9|a^2=d,b^2=ei,c^2=fg,d^2=efg,f^2=gh,g^2=hi,h^2=i,b^a=bc,c^a=ce,c^b=cf,d^b=def,f^b=fg,g^b=gh,h^b=hi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1937);